## Abstract

We investigate the non-equilibrium dynamics of an ordered stripe-forming system free of topological defects. In particular, we study the ageing and the coarsening of orientation fluctuations parallel and perpendicular to the stripes via computer simulations based on a minimal phase-field model (model B with Coulomb interactions). Under the influence of noise, the stripe orientation field develops fluctuations parallel to the stripes, with the dominant modulation length λ*_{∥} increasing with time *t* as λ*_{∥} ∼ *t*^{1/4} and the correlation length perpendicular to the stripes *ξ*^{⊥}_{θ} increasing as *ξ*^{⊥}_{θ} ∼ *t*^{1/2}. We explain these anisotropic coarsening dynamics with an analytic theory based on the linear elastic model for stripe displacements first introduced by Landau and Peierls. We thus obtain the scaling forms and the scaling exponents characterizing the correlation functions and the structure factor of the stripe orientation field. Our results reveal how the coarsening of orientation fluctuations prevents a periodically modulated phase free of topological defects from reaching equilibrium.

## 1. Introduction

A periodic stripe pattern is a conceptually simple prototype of more complex modulated phases, such as the periodic patterns of lamellae, hexagonally ordered cylinders and spheres, as well as the gyroid phase [1]. Periodically modulated phases are found in a multitude of physical systems [2], for instance, block copolymers [3], liquid crystals [4], Rayleigh–Bénard convection [5,6], thin magnetic films [7–10], electronic phases in superconductors [11,12] and lipid membranes [13]. Shape and orientation fluctuations, due to either thermal noise or a stochastically fluctuating external parameter, are an intrinsic property of any modulated phase. Shape and orientation fluctuations are also essential for enabling the motion of point and line defects in periodically modulated phases, which is the fundamental mechanism required for defect motion, long-range ordering and ductility.

Our initial motivation for studying stripe phases was to quantify the microdomain dynamics in thin films of cylinder-forming block copolymers. The motion of point defects [14–19], the structural phase transformations between different types of microdomain patterns [20], as well as the shape and orientation fluctuations of individual block copolymer microdomains [21,22] can be imaged with high spatial and temporal resolution using atomic force microscopy (AFM). This results in a time series of two-dimensional images of the microdomain pattern at the film surface. Unfortunately, AFM images are often afflicted with imaging artefacts, in particular, shot noise, temporal contrast variations, and spatial drift that precludes imaging a fixed area over a long time period. To avoid these problems, we decided to use a minimal phase-field model describing the block copolymer microdomain dynamics (model B with Coulomb interactions [23,24]) to create stripe patterns without imaging artefacts. This model has been used for studying the microphase separation in block copolymers [24–30] as well as the applications of block copolymers as lithographic masks [31–35]. The model also describes reactive binary mixtures [36,37]. The underlying free-energy term has also received considerable attention for describing the magnetic domain pattern of Coulomb-frustrated ferromagnets [38–40] and for possible scenarios for the glass transition [41–44]. Other types of interactions are likewise known to cause the emergence of stripe phases [45]. The minimal phase-field model is computationally much less demanding than the more sophisticated models of block copolymer microdomain dynamics [46,47], allowing us to more easily study the long-term dynamics of stripe phases.

Since the simulation results are also a time series of two-dimensional images, we use quantitative image analysis to extract physical parameters from the stripe patterns. With this approach, we discovered slow non-equilibrium dynamics and ageing of orientation fluctuations in a stripe phase without topological defects that evolves under the influence of noise [48].

Two important phenomena of non-equilibrium dynamics are ageing and coarsening [49,50]. A system exhibiting temporal fluctuations is in a stationary state when its dynamics is translationally invariant in time. The two-time correlation function *C*(*t*, *t*_{w}) ≡ 〈*Y* (*t*)*Y*(*t*_{w})〉 − 〈*Y*(*t*)〉〈*Y* (*t*_{w})〉 of an observable *Y*(*t*), where 〈 · 〉 denotes a statistical average, allows for a rigorous test. Here, *t* is the observation time and *t*_{w} < *t* denotes the waiting time. In the stationary state, *C*(*t*, *t*_{w}) is translationally invariant in time and depends only on the difference *t* − *t*_{w}. By contrast, if *C*(*t*, *t*_{w}) depends explicitly on both arguments, the system is not in a stationary state and therefore not in equilibrium. An example is ageing, where the dynamics slows down continuously as *t*_{w} is increased, and the system never reaches a stationary state. Ageing processes have been investigated extensively in glasses [51], which exhibit a dramatic slow-down of atomic and molecular motion during cool-down towards and below the glass transition temperature. However, ageing processes have also been investigated in other non-equilibrium systems [50]. Phenomenologically, the ageing behaviour of many systems can be captured by a scaling form for the two-time correlation function referred to as simple ageing [50]:
1.1where *b* is the ageing exponent and *f*(*x*) a scaling function.

Ageing often concurs with spatial coarsening [52–54], for example, in phase-ordering systems, such as magnets and binary mixtures, where the dynamics is driven by domain growth and the annihilation of topological defects [49,55]. Coarsening and ageing are also observed in surface roughening phenomena, where height fluctuations grow under the influence of noise [50,56,57]. Stripe phases also display ageing and coarsening after a quench from a homogeneous state to a modulated (ordered) state. In this process, domains with long-range order (grains) grow via the annihilation of defects, such as dislocations and grain boundaries between domains with different lattice orientation [58–68]. During domain growth, the defect-annihilation rate slows down continuously as the average distance between dislocations and between grain boundaries increases [69,70].

In our previous work, we used computer simulations based on model B with Coulomb interactions to study orientation fluctuations in stripe phases free of topological defects [48]. The model consists of a stochastic partial differential equation (SPDE) for describing the temporal evolution of a concentration field *ψ*(** r**,

*t*) under the influence of noise of strength

*η*(see §2.1, equation (2.1)). In our case,

*η*is well below the critical noise strength

*η*

_{c}. An example of the resulting stripe pattern is shown in figure 1

*a*. A perfect stripe pattern with the stripe period λ

_{0}that minimizes the system's free energy was used as the initial condition of the simulation. Subsequently, the stripes evolved under the influence of noise. The stripe centre lines can be described with the local stripe displacement

*u*(

**,**

*r**t*) perpendicular to the stripe direction and the local stripe orientation

*θ*(

**,**

*r**t*), as shown in figure 1

*b*.

In [48], we analysed the two-time orientation correlation function
1.2where 〈 · 〉_{r,ζ} denotes an average over positions ** r** ≡ (

*r*

_{⊥},

*r*

_{∥}) and realizations of the noise

*ζ*(

**,**

*r**t*). In the ordered state, where stripes with long-range order exist,

*C*

_{θ}(

*t*,

*t*

_{w}) lacks time-translational invariance and depends on the waiting time

*t*

_{w}when plotted as a function of

*t*−

*t*

_{w}. The slow-down with increasing waiting time can be described by the scaling form, equation (1.1), typical for ageing. This shows that the system is not in equilibrium, despite the long-range order of the stripes and despite the absence of topological defects. This is very astonishing since an ordered configuration is generally closer to equilibrium than a disordered configuration. The ageing concurs with a spatial coarsening of orientation fluctuations that is visible in the maps of the stripe orientation field

*θ*(

**,**

*r**t*) for increasing simulation time

*t*(figure 2

*a*–

*c*). Evolving from perfectly ordered stripes at

*t*= 0, the pattern of the orientation fluctuations is fine-grained at first and coarsens with time. This coarsening is reflected in the growth of the characteristic correlation length

*ξ*

^{⊥}

_{θ}in the direction perpendicular to the stripes [48] and in the increase of the dominant modulation length λ*

_{∥}along the stripe direction (this work). The spatial coarsening of the orientation fluctuations is more easily recognized in the snapshots of the orientation fields where fluctuations smaller than λ

_{0}were removed using a Gaussian filter (figure 2

*d*–

*f*). The superimposed streamlines give an impression of the local stripe displacement (multiplied by a factor of 20).

Stripe orientation patterns very similar to those shown in figure 2 have been observed in thin films of cylinder-forming block copolymers [18,19,71–73]. An example is shown in figure 3. The data are taken from Hammond & Kramer [72], where a thin film of a cylinder-forming block copolymer was deposited in a shallow trough with fixed side walls. The walls induce the preferential orientation of the monolayer of cylinders (stripes); however, at a distance sufficiently far from the side walls, small-angle orientation variations still exist, as figure 3 shows. In nanotechnology, there is a large interest in using block copolymer self-assembly to obtain stripe and other types of patterns that can be used as lithographic masks [74], as templates for bit-patterned magnetic media [75,76] and as photonic crystals [77,78]. To this end, film–substrate interfaces [79], periodically structured substrates [80], electric fields [81], shearing [82,83] and zone casting [84,85] can be used for directing the self-assembly and for reducing the number of defects within the pattern (for recent reviews, see [86–88]). In this context, the defect annihilation dynamics and the coarsening of orientational domains have been studied in block copolymer films forming stripe patterns [14–19,89–92]. The intrinsic dynamics of stripe orientation fluctuations is apparent in time-lapse image series of stripe patterns observed in the thin films of cylinder-forming block copolymers [21,22]. However, imaging artefacts and the limited time window of experimental data have so far precluded the quantitative statistical analysis of experimental data on the spatio-temporal dynamics of the orientation fluctuations in stripe phases.

The purpose of this work is to derive the orientation correlation functions and structure factors as well as their scaling properties based on the linear elastic model for stripe displacements first introduced by Landau & Peierls [93,94]. This theory explains the ageing dynamics and the spatial coarsening of orientation fluctuations in stripe phases in great detail, which we discovered on the basis of computer simulations [48]. The linear elastic model is generic, since it phenomenologically describes small-angle orientation fluctuations of many stripe-forming systems based on their symmetry. Therefore, our results have important implications for understanding the thermodynamics of stripe phases in nature.

## 2. Models and observables

### 2.1. Phase-field model for stripe dynamics

We have performed numerical simulations of a minimal model for stripe formation and dynamics, known as model B with Coulomb interactions. It is based on the dynamic equation for phase separation with a conserved order parameter [95]:
2.1where *ψ*(**r**, *t*) is a scalar field representing the concentration difference *ψ*(**r**, *t*) = *ψ*_{A}(**r**, *t*) − *ψ*_{B}(**r**, *t*) between two components *A* and *B*. The Gaussian noise term *ζ*(**r**, *t*) satisfies 〈*ζ*(**r**, *t*)〉 = 0 and 〈*ζ*(**r**, *t*)*ζ*(**r**′, *t*′)〉 = − 2*η*∇^{2}*δ*(**r** − **r**′)*δ*(*t* − *t*′), where 〈 · 〉 stands for the statistical average and *η* parametrizes the noise strength. The free-energy functional was first derived by Ohta & Kawasaki [23]. In *d* spatial dimensions,
2.2where the first integral is a short-range Ginzburg–Landau free energy, and the second integral represents repulsive long-range Coulomb interactions controlled by the parameter *Γ*. Green's function *G*(**r** − **r**′) is defined by −∇^{2}*G*(**r** − **r**′) = *δ*(**r** − **r**′). Inserting the free energy, equation (2.2), into the dynamic equation, equation (2.1), yields the following SPDE:
2.3Compared to more sophisticated approaches for simulating block copolymers [46,47], the model given by equation (2.3) has the advantage of being computationally less demanding, which enables us to more easily investigate its long-term behaviour.

As detailed in our previous work [48], we prepare the system in a perfectly ordered state consisting of parallel stripes with wavelength λ_{0} = 2*π**Γ*^{−1/4}. The amplitude and wavelength were chosen to minimize the free energy in a single-mode approximation [37]. We fix the interaction parameter *Γ* as 0.2, for which equation (2.3) exhibits stripe formation [63] with λ_{0} ≈ 9.3956. The noise strength *η* was varied, but remains below the critical noise strength *η*_{c} = 0.020(3), at which long-range order in the stripe pattern is destroyed [48]. Using an efficient pseudo-spectral algorithm [96], we performed simulations of equation (2.3) with periodic boundary conditions in *d* = 2 dimensions on lattices with size *L* = 55λ_{0}. Space and time were discretized in increments of Δ*r* = 1 (Δ*r* = λ_{0}/10 in §3.2) and Δ*t* = 0.1, respectively. The results presented below were averaged over 40 independent realizations. A subset of the data is from [48].

### 2.2. Correlation functions

Our main observable is the local stripe orientation *θ*(**r**, *t*), computed from the concentration field *ψ*(**r**, *t*) using the gradient-square tensor [97,98], also referred to as the structure tensor [99]:
2.4The operator ∂_{⊥} (∂_{∥}) represents the partial derivative with respect to *r*_{⊥} (*r*_{∥}). The local stripe orientation *θ*(**r**, *t*) is given by the direction of the eigenvector of **G**(**r**, *t*) associated with the largest eigenvalue. We used the algorithm described in [100] for the numerical calculation of the orientation.

To investigate the dynamics of the stripe orientation *θ*(**r**, *t*), we use spatio-temporal correlation functions. The most general form is given by
2.5which takes into account the symmetry and periodicity of the stripe orientation, namely, its invariance under the transformation *θ* → *θ* + *π*. In the case of spatial translation invariance, equation (2.5) depends only on the difference **R** ≡ **r** − **r**′ and can thus be written as *C*_{θ}(**R**, *t*, *t*_{w})≡*C*_{θ}(**r**, **r** + **R**, *t*, *t*_{w}). From this quantity, the two-time correlation function *C*_{θ}(*t*, *t*_{w}) is obtained by setting **R** = 0:
2.6Conversely, the equal-time spatial correlation function is given by
2.7We also consider a related quantity, namely, the structure factor of the orientation, *S*_{θ}(**q**,*t*), which is given by the Fourier transform of the spatial orientation correlation function:
2.8For small angles *θ*, such as those observed at small noise strengths *η* ≪*η*_{c}, the orientation correlation function, equation (2.5), simplifies to *C*_{θ}(**r**, **r**′,*t*, *t*_{w}) ≃ 4[〈*θ*(**r**, *t*)*θ*(**r**′, *t*_{w})〉_{ζ} − 〈*θ*(**r**, *t*)〉_{ζ}〈*θ*(**r**′,*t*_{w})〉_{ζ}]. We will use this approximation in our analytical calculations in §3.

### 2.3. Landau–Peierls model for stripe displacement

In this section, we discuss a model for smectic liquid crystals [4] in *d* = 3 dimensions and stripe-forming systems (*d* = 2), which dates back to works by Peierls [93] and Landau [94]. Later, the theory was developed by de Gennes [101] for smectic liquid crystals and investigated in two dimensions also for stripe phases by Toner & Nelson [102]. In particular, the static structure factor and the spatial correlation function of the displacement field were computed [102,103]. The model is defined in terms of a displacement field *u*(** r**,

*t*), which indicates the distance between a stripe (a layer in

*d*= 3) and its unperturbed position (figure 1

*b*). It has been shown that both smectics and stripe-forming systems can be described by the free-energy functional [59,102,104] 2.9where

*u*(

**,**

*r**t*) represents the displacement field and

*ν*

_{2},

*ν*

_{4}> 0 are elastic constants related to compression and bending, respectively. The operator ∂

_{⊥}(∂

_{∥}) represents the partial derivative with respect to

*r*

_{⊥}(

*r*

_{∥}). In the case of the stripe-forming system given by equation (2.3), the elastic constants take on the values and

*ν*

_{4}= 1 [104,105]. We now consider the SPDE 2.10with the correlations of the white noise

*ζ*

_{u}(

**r**,

*t*) given by 〈

*ζ*

_{u}(

**r**,

*t*)

*ζ*

_{u}(

**r**′,

*t*′)〉 = 2

*σ*

^{2}

*δ*(

**r**−

**r**′)

*δ*(

*t*−

*t*′), where

*σ*

^{2}is the noise strength. Equation (2.10) represents a gradient descent dynamics within the energy combined with the stochastic noise term

*ζ*

_{u}(

**r**,

*t*) and thus corresponds to model A in the classification of Hohenberg & Halperin [95]. After introducing the Fourier transform , with

**q**≡(

*q*

_{⊥},

*q*

_{∥}), the equation reads 2.11where the Fourier-transformed noise has correlations . Since we are interested in the dynamics of the orientation field

*θ*(

**r**,

*t*), we follow Toner & Nelson [102] and compute

*θ*(

**r**,

*t*) from the displacement field

*u*(

**r**,

*t*) by taking the derivative in the

*r*

_{∥}direction (figure 1

*b*):

*θ*(

**r**,

*t*)≡ − tan

^{−1}[∂

_{∥}

*u*(

**r**,

*t*)] ≃−∂

_{∥}

*u*(

**r**,

*t*), where the latter approximation holds for small gradients of the displacement ∂

_{∥}

*u*(

**r**,

*t*)≪1. Applying this to equation (2.11) leads to the following equation for the Fourier transform of the orientation field

*θ*(

**q**,

*t*): 2.12where the noise

*ζ*

_{θ}has correlations 〈

*ζ*

_{θ}(

**r**,

*t*)

*ζ*

_{θ}(

**r**′,

*t*′)〉 =−2

*σ*

^{2}∂

^{2}

_{∥}

*δ*(

**r**−

**r**′)

*δ*(

*t*−

*t*′) in real space, which corresponds to in Fourier space. Equation (2.12) is a linear stochastic differential equation (SDE), similar to those arising in the context of surface roughening processes [57], which are known to exhibit coarsening and ageing [50].

## 3. Results

### 3.1. Ageing

#### 3.1.1. The two-time correlation function *C*_{θ}(*t*, *t*_{w})

The solution of equation (2.12) can be written as , with *a*(**q**)≡*ν*_{2}*q*^{2}_{⊥} + *ν*_{4}*q*^{4}_{∥}, for the initial condition . Using this expression, we define the following correlation function:
3.1which can be evaluated as
3.2
3.3
3.4where the waiting time *t*_{w} ≤ *t*. We note that the average vanishes for all times *t*. By using the spatial translation invariance of the correlation function *C*_{θ}(**r**, **r** + **R**, *t*, *t*_{w}) and applying the Wiener–Khintchine theorem [106], the following identity is obtained:
3.5where is the two-time structure factor. A comparison of equations (3.2) and (3.3) yields
3.6From this expression, the corresponding two-time correlation function *C*_{θ}(*t*,*t*_{w}) can be obtained by integrating over all wavenumbers **q**:
3.7
3.8This function admits the scaling form *C*_{θ}(*t*, *t*_{w}) ∼ *t*^{−1/4}_{w}*f*(*t*/*t*_{w}), which yields the ageing exponent . The scaling function is given by *f*(*x*) = (*x* − 1)^{−1/4} − (*x* + 1)^{−1/4}.

In figure 4, equation (3.8) is plotted as a function of the time difference *t* − *t*_{w} for a wide range of waiting times *t*_{w}. The separation of time scales, a hallmark of ageing systems [49,51], becomes readily apparent. For *t* − *t*_{w} ≪ *t*_{w}, there is a regime where equation (3.8) depends only on the difference *t* − *t*_{w}: *C*_{θ}(*t*, *t*_{w}) ∼ (*t* − *t*_{w})^{−1/4}. This quasi-stationary dynamics takes place on short time scales, where the system does not respond to its age *t*_{w}. A second regime is observed when *t* − *t*_{w} ≫ *t*_{w}, resulting in *C*_{θ}(*t*, *t*_{w}) ∼ *t*_{w}(*t* − *t*_{w})^{−5/4}: the system exhibits ageing. Also shown in figure 4 are numerical data from the stripe-forming system studied in [48], which are located mostly in the cross-over region between the two limiting behaviours exhibited by equation (3.8). All in all, theory and the numerical data agree excellently over the entire range of simulation time.

#### 3.1.2. Scaling behaviour of *C*_{θ}(*t*, *t*_{w})

We proceed by investigating the scaling behaviour of *C*_{θ}(*t*, *t*_{w}). In figure 5, we plotted *C*_{θ}(*t*, *t*_{w}) for fixed ratios *t*/*t*_{w} in order to determine the exponent *b* in the prefactor *t*^{−b}_{w}. *C*_{θ}(*t*, *t*_{w}) from equation (3.8), which decays as *t*^{−1/4}_{w} in this representation, is shown for comparison. For each of the values *t*/*t*_{w}, there is excellent agreement between the numerical results and the theoretical predictions, considering the fluctuations present in the numerical data. Only for very small waiting times *t*_{w} are small but systematic deviations visible. In [48], we reported that the exponent *b* depends weakly on the noise strength, increasing from *b* = 0.22 for to *b* = 0.33 for . The theoretical result, equation (3.8), predicts independent of the noise strength. However, the numerical values for *b* given in [48] were obtained by fitting a restricted range of data exhibiting substantial fluctuations. Numerical studies of larger systems observed for longer times and a larger number of realizations would be necessary to determine the value of the exponent at different noise strengths *η* with higher accuracy and for identifying possible finite-size effects.

Finally, we test the scaling properties of the orientation correlation function *C*_{θ}(*t*, *t*_{w}) by multiplying the numerical data by *t*^{1/4}_{w} and plotting them as a function of (*t* − *t*_{w})/*t*_{w} (figure 6). For a wide range of noise strengths *η*, the data points collapse onto individual master curves. These curves are well described by the scaling function *f*(*t*/*t*_{w}) obtained from equation (3.8), with the prefactor adjusted proportional to the noise strength. The collapse is especially good at small values of *t*/*t*_{w}, whereas the scatter becomes larger as *t*/*t*_{w} increases. Qualitatively, the collapse obtained using is as convincing as the one presented in [48], where slightly different exponents *b* were used for each noise strength. We also emphasize the asymptotic behaviour of the scaling function for *t*/*t*_{w} → ∞, where *f*(*t*/*t*_{w}) decays as a power law ∼(*t*/*t*_{w})^{−5/4}. This behaviour is used to define the autocorrelation exponent [50,107] . The corresponding power law is plotted as a black line in figure 6. While the numerical data are consistent with a power-law decay at large *t*/*t*_{w}, there is significant scatter in the range (*t* − *t*_{w})/*t*_{w} ≳ 10, where the power law becomes apparent. This is due to the small magnitude of the correlation function *C*_{θ}(*t*, *t*_{w}), which is therefore more susceptible to statistical fluctuations.

### 3.2. Anisotropic coarsening of orientation fluctuations

#### 3.2.1. Coarsening parallel to the stripes

We first address the dynamics described by equation (2.12) in the *r*_{∥}-direction using the equal-time structure factor *S*_{θ}(**q**, *t*)≡*S*_{θ}(**q**, *t*, *t*_{w} = *t*) obtained from equation (3.4). A cut through the structure factor in the *q*_{∥} direction yields
3.9This function has a maximum at a wavenumber , where the constant and *W*_{−1}( · ) represents the negative branch of the Lambert *W* function [108]. The wavenumber *q**_{∥}(*t*) corresponds to the wavelength λ*_{∥}(*t*)≡2*π*/*q**_{∥}(*t*) ∼ *t*^{1/4}, which we will refer to as the dominant modulation length. The structure factor given by equation (3.9) can be written in scaling form: *S*_{θ}(*q*_{⊥} = 0, *q*_{∥}, *t*) ∼ *f*[*q*_{∥}/*q**_{∥}(*t*)]/*q**_{∥}(*t*)^{2}, indicating that the intensity at the maximum increases with time: , with the constant *c*_{1}≡(1 − e^{−c0})^{−1} ≈ 1.3986.

In the following, we compare these theoretical results with numerical data for the stripe-forming system. As shown in figure 7*a*, the cuts *S*_{θ}(*q*_{⊥} = 0, *q*_{∥}, *t*) parallel to the stripes exhibit a single peak, the intensity of which increases with time. Simultaneously, the peak position shifts to lower *q*_{∥}. At large wavenumbers, the structure factor decreases rapidly, independent of time. Comparing the data for different times *t*, we note that there is no evolution for *q*_{∥}/*q*_{0} ≳1, that is, the curves for different times coincide with each other. This range corresponds to small distances (<λ_{0}), where the linear elastic model, equation (2.10), is not expected to adequately describe the dynamics due to the coarse-graining inherent in its derivation [104]. Furthermore, the algorithm [100] used to extract the stripe orientation acts as a filter that reduces the amplitude of modes with short wavelengths. The latter is the main reason for the decrease of the structure factor for *q*_{∥}/*q*_{0} ≳1. As time progresses, the structure factor becomes static over an increasing range of wavenumbers *q*_{∥}/*q*_{0} < 1. To extract the peak position *q*_{∥}* and intensity *S**_{θ}, we fitted the theoretical structure factor given by equation (3.9) to the numerical data. As shown in the inset in figure 7*a*, there is excellent agreement between theory and the numerical results. The dominant modulation length λ_{∥}* obtained from the fitting procedure is plotted as a function of time in figure 8. It increases as a power law ∼*t*^{1/4}, with the exponent as predicted by equation (3.9). We also note that λ*_{∥}(*t*) does not attain stationarity within the time range considered here, indicating that the coarsening process is still in progress at the latest times examined. The peak intensity *S**_{θ}(*t*) increases as a power law ∼*t*^{1/2} (see inset in figure 7*b*), also in agreement with equation (3.9).

To further illustrate the scaling behaviour of the structure factor, we rewrite *S*_{θ}(*q*_{⊥} = 0, *q*_{∥}, *t*) in a rescaled form. Introducing a reduced wavenumber, , and dividing by the peak intensity *S**_{θ}(*t*), the rescaled structure factor reads
3.10The structure factor is now independent of any model-specific parameters, which might be useful for comparison with experimental data, and the time dependence has been absorbed into the quantity . Equation (3.10) is plotted as red line in figure 7*b*. It exhibits a characteristic wedge-like shape formed by power laws for and for . The structure factor determined from numerical simulations *S*_{θ}(*q*_{⊥} = 0, *q*_{∥}, *t*), rescaled according to equation (3.10), is also plotted in figure 7*b* for different times *t*. For , the numerical data for different times collapse onto a master curve, which is well described by equation (3.10). As time progresses, more and more data points lie on the slope . As *t* → ∞, this decay dominates the behaviour of the structure factor in a finite system [109]. For large values of , the structure factor decreases rapidly and does not exhibit scaling behaviour, due to the filtering effect mentioned above.

#### 3.2.2. Coarsening perpendicular to the stripes

We now turn to the coarsening dynamics of orientation fluctuations perpendicular to the stripes [48]. The equal-time spatial correlation function *C*_{θ}(*r*_{⊥}, *r*_{∥} = 0, *t*) represents a cut through *C*_{θ}(**R**, *t*) (see equation (2.7)) perpendicular to the stripes. It can be computed from the equal-time structure factor *S*_{θ}(**q**, *t*):
3.11
3.12where represents the incomplete Gamma function [110]. Equation (3.12) is a product of two terms which depend on the distance *r*_{⊥}: the incomplete Gamma function in the numerator, and the square root in the denominator. The Gamma function depends on the argument (*r*_{⊥}/*ξ*^{⊥}_{θ})^{2}, where we have identified the correlation length . Thus, the correlation function can be written in scaling form: . We note that equation (3.12) exhibits a transition from rapid decay to power-law behaviour as *ξ*^{⊥}_{θ} increases, mediated by the properties of the Gamma function for large and small arguments. This behaviour will be explored below, in conjunction with a comparison to numerical data from the stripe-forming system. In figure 9*a*, we show the spatial correlation function *C*_{θ}(*r*_{⊥}, *r*_{∥} = 0, *t*) at different times *t* (red lines), along with the numerical results computed according to equation (2.7) (green). Equation (3.12) has been fitted to the numerical data, with the correlation length *ξ*^{⊥}_{θ} as a fit parameter. For all but the latest times, the fit function describes the data very well. The correlation length *ξ*^{⊥}_{θ} increases with time as a power law *ξ*^{⊥}_{θ} ∼ *t*^{1/2}, as predicted by equation (3.12), and reaches the system size at *t* ≈ 10^{4} (figure 8). Besides the growing correlation length, the shape of the correlation function also changes over time. At early times, *C*_{θ}(*r*_{⊥}, *r*_{∥} = 0, *t*) decays within a short distance, indicating a small correlation length. For times *t* ≳ 5 × 10^{4}, the curve resembles a power law, as reported earlier [48].

Similar to the structure factor discussed in the preceding section, *C*_{θ}(*r*_{⊥}, *r*_{∥} = 0, *t*) can be cast into a time-independent form by introducing a reduced distance . The rescaled correlation function is given by
3.13This function depends only implicitly on time through the quantity . To test the scaling behaviour, we rescaled *C*_{θ}(*r*_{⊥}, *r*_{∥} = 0, *t*), which was computed from the stripe-forming system according to equation (3.13). In figure 9*b*, the numerical data and the theoretical curve are plotted as a function of the reduced distance . At early and intermediate times, we observe a convincing collapse of the data. As time progresses (and therefore decreases), the data points move from left to right. Two distinct regimes can be identified: For , the numerator in equation (3.13) decays rapidly, resulting in (see the dotted curve in figure 9*b*). On the other hand, for , the incomplete Gamma function in the numerator approaches the constant value . Thus, behaves like a power law . Only at *t* = 5 × 10^{5} is there a mismatch between theory and the numerical data. The correlation function *C*_{θ}(*r*_{⊥}, *r*_{∥} = 0, *t*) obtained from the stripe-forming system decays as a power law ∼*r*^{−c}_{⊥}, with *c* ≈ 0.33 [48], whereas theory predicts . Simulations of smaller systems (data not shown) indicate that the observed value of the exponent *c* depends on the system size *L*. Therefore, we interpret this deviation between theory and the numerical results as a finite-size effect.

## 4. Discussion

The ageing and coarsening dynamics of small-angle orientation fluctuations in stripe phases free of topological defects are accounted for by a reduced model based on the linear elastic theory for the stripe displacement field known as the Landau–Peierls model [4]. This model is generic, since it phenomenologically describes small-angle orientation fluctuations of many stripe-forming systems. Our numerical simulations [48] based on model B with Coulomb interactions agree excellently with this theory. The two-time correlation function *C*_{θ}(*t*, *t*_{w}) displays a characteristic slow-down with increasing waiting time *t*_{w}, and it obeys a scaling form that is characteristic for simple ageing with the ageing exponent and the autocorrelation exponent . The continuous slow-down and lack of time-translational invariance show that the system is not in equilibrium. The linear elastic theory based on the Landau–Peierls model shows that this ageing and non-equilibrium dynamics is an intrinsic property of many stripe-forming systems and that it occurs for the smallest amount of noise. The ageing concurs with spatial coarsening and dynamical scaling as in many other systems displaying ageing [54]. The dominant modulation length λ*_{∥} parallel to the stripes increases as *t*^{1/4} and the correlation length *ξ*^{⊥}_{θ} perpendicular to the stripes increases as *t*^{1/2}. This anisotropic coarsening is also described with the Landau–Peierls model. As a result, the orientation field *θ*(**r**, *t*) forms a spatially complex pattern that is continuously changing. The pattern coarsens with different growth velocities parallel and perpendicular to the stripes, and this growth is continuously slowing down. Such a continuous change of the spatial pattern and its dynamics (the temporal pattern) are hallmarks of an ageing system that is not in equilibrium.

Note that this slow dynamics is not visible in the temporal evolution of the free-energy density for the phase-field model given by equation (2.3), which appears to become stationary for *t* > 100 (see fig. 1*e* in [48]). From the orientation field *θ*(**r**, *t*) computed from our phase-field simulations for , we determined the ratio of the two contributions to the Landau–Peierls free energy functional (equation (2.9)), which correspond to compression and bending, respectively. Averaging over 40 noise realizations, we find that the ratio does not change (data not shown). This shows that (on average) the energetic changes associated with the pattern's coarsening are much smaller than the energetic fluctuations due to the noise.

In numerical simulations, the coarsening is limited by the system size. This finite-size effect and the coupling of the coarsening dynamics parallel and perpendicular to the stripes will be the topic of a forthcoming publication. The spatial orientation correlation function perpendicular to the stripes *C*_{θ}(*r*_{⊥}, *r*_{∥} = 0, *t*) decays as a power of *r*_{⊥} at late times. Such a behaviour can be found at a critical point [111] or in a critical phase [112], but also in self-organized criticality [113], where power-law correlations emerge naturally from the intrinsic dynamics of the system.

The roughening of surfaces can display ageing and coarsening that is in many aspects similar to the ageing and coarsening of orientation fluctuations in stripe phases. For example, the scaling form of the two-time correlation function *C*_{θ}(*t*, *t*_{w}), equation (3.8), has the same form as the two-time correlation function found for certain surface roughening phenomena [114]. The SPDE (2.10) describing the dynamics of the displacement field *u*(**r**, *t*) can be considered as a combination of the Edwards–Wilkinson [115] (EW) equation and the noisy Mullins–Herring [116] (MH) equation. Setting *ν*_{4} = 0 reduces equation (2.10) to the one-dimensional EW equation. Setting *ν*_{2} = 0 reduces equation (2.10) to the noisy, one-dimensional MH equation.

In the direction parallel to the stripes, the SDE (2.12) for the stripe orientation *θ*(**r**, *t*) resembles (in Fourier space) the one-dimensional MH equation with conserved noise (MHc) that is sometimes referred to as the LCC4 model [56]. This resemblance is evidenced by the structure factor *S*_{θ}(*q*_{⊥} = 0, *q*_{∥}, *t*), which is identical to the one obtained for the MHc equation [109,117,118]. The corresponding universality class is characterized by a set of scaling exponents: the roughness exponent , the growth exponent and the dynamic exponent *z* = 4 [56]. Using these exponents, the structure factor can be written in a scaling form corresponding to the Family–Vicsek scaling [119]: , where the scaling function decays asymptotically as *x*^{−2α−1} for *x* → ∞. In a similar vein, we can determine the scaling exponents for the correlation function perpendicular to the stripes *C*_{θ}(*r*_{⊥}, *r*_{∥} = 0, *t*)∼*t*^{2β}*f*(*r*_{⊥}/*t*^{1/z}) with , *z* = 2 and . These exponents do not represent a universality class listed in [56]. The inherent anisotropy of a stripe phase and of the Landau–Peierls model describing small stripe displacements causes an interesting combination of two universality classes: one for the coarsening parallel to the stripes and the other for the coarsening perpendicular to the stripes.

The analogy with surface roughening gives insight into the mechanism causing the coarsening and ageing of orientation fluctuations in stripe phases. In the Landau–Peierls model, the noise term *ζ*_{u}(**r**, *t*) in the SPDE (2.10) describing the dynamics of the stripe displacement field *u*(**r**, *t*) causes a continuous stochastic perturbation of the stripe displacement. Displacements with a large local curvature (short wavelength) are energetically unfavourable and decay in favour of stripe deformations with less local curvature (long wavelength). As the noise term is spatially and temporally homogeneous, this causes a continuous spatial coarsening. The intrinsic spatial anisotropy of the stripe phase leads to the spatially anisotropic coarsening and ageing, with different universality classes in the direction parallel to the stripes and in the direction perpendicular to them.

At last, we mention an important difference between the orientation fluctuations described by equation (2.12) and the height fluctuations observed in models of surface roughening, such as the EW equation, the MH equation, and others [56,57]. In the latter class of models, the roughness diverges with the system size. In the Landau–Peierls model, however, the analogous quantity 〈*θ*(**r**, *t*)^{2}〉_{ζ,r} remains finite, as shown by Toner & Nelson [102], meaning that stripe phases described by equation (2.12) retain orientational order.

We expect that the presented methodology for characterizing the spatio-temporal dynamics of stripe patterns will be also useful for analysing the dynamics of stripe patterns observed in experiments. An example is the orientation field of the stripe pattern formed in a thin film of cylinder-forming block copolymers shown in figure 3. Hammond & Kramer [72] have not studied the long-term dynamics of these orientation fluctuations. Therefore, we do not know whether this particular stripe pattern displays ageing. Nevertheless, the similarity with the snapshots of the orientation fields from our numerical simulations is striking.

## 5. Conclusion

Ageing and anisotropic spatial coarsening of orientation fluctuations prevent stripe phases free of topological defects from reaching equilibrium. This result has important implications for understanding the thermodynamics of stripe phases and probably also other periodically modulated phases. In particular, it raises the question of whether stripe phases can reach equilibrium and how they might do so. Furthermore, the spatial coarsening of orientation fluctuations poses an intrinsic and fundamental limit for the perfection of self-assembled or self-organized periodic stripe patterns. Some interesting questions for future research are the properties of orientation fluctuations in other periodically modulated phases and how periodically patterned substrates, external fields and boundary conditions at external walls influence the dynamics of small-angle orientation fluctuations.

## Authors' contributions

The authors jointly planned the study, interpreted the results and developed the analytical theory. C.R. implemented and analysed the numerical simulations and performed the analytical calculations. C.R. and R.M. drafted the paper; G.R. provided critical revisions.

## Competing interests

We declare we have no competing interests.

## Funding

We received no funding for this study.

## Footnotes

One contribution of 17 to a theme issue ‘Growth and function of complex forms in biological tissue and synthetic self-assembly’.

- © 2017 The Author(s)

Published by the Royal Society. All rights reserved.