Marianne Duyck

Thérèse Collins

Mark Wexler
email: mark.wexler@u-paris.fr

Visual continuity during blinks and alterations in time perception

Keywords: blinks, time perception, vision

Eye blinks strongly attenuate visual input, yet we perceive the world as continuous. How this visual continuity is achieved remains a fundamental and unsolved problem. A decrease in luminance sensitivity has been proposed as a mechanism, but is insufficient to mask the even larger decrease in luminance due to blinks. Here we put forward a different hypothesis: visual continuity can be achieved through shortening of perceived durations of the sensory consequences of blinks. Here we probed the perceived durations of the black-outs caused by blinks, and of visual stimuli interrupted by blinks. We found that the perceived durations of blackouts due to blinks are about half as long as artificial black-outs immediately preceding or following the blink. Stimuli interrupted by blinks were perceived as briefer than uninterrupted stimuli, by about the same duration as the interruption-but so were stimuli interrupted by optically simulated blinks. There was a difference between real and simulated blinks, however: the decrease in perceived duration depended on the duration of the interruption for simulated, but not for real, blinks. These profound modifications in time perception during blinks show a way in which temporal processing contributes to the solution of an essential perceptual problem.

Significance statement

Although eye blinks effectively shut off the light entering the eyes for brief but significant periods every few seconds, we hardly perceive the black-outs or the visual images seen before and after the blink as interrupted. In this study we showed that the perception of time is altered around blinks: both the period of darkness during the blink, and the duration of brief images straddling the blink, are perceived as significantly briefer than equivalent periods of darkness before or after the blink, or images with the same duration that aren't interrupted by blinks. This alteration in the perception of time may be the reason why we hardly perceive the interruptions in vision due to blinks.

Introduction

We blink on average 15 times per minute [START_REF] Ponder | On the act of blinking[END_REF][START_REF] Bentivoglio | Analysis of blink rate patterns in normal subjects[END_REF].

Blinks cause a drastic decrease in luminance-about 1.5-2.5 log units, depending on frequency [START_REF] Ando | Light attenuation by the human eyelid[END_REF][START_REF] Moseley | Light transmission through the human eyelid: In vivo measurement[END_REF][START_REF] Robinson | Transmission of light across the adult and neonatal eyelid in vivo[END_REF]-and a near-complete loss of pattern vision, typically lasting about 100-300 ms [START_REF] Slater-Hammel | Blackout Interval during Eye Blinks[END_REF]. However, these drastic sensory consequences of blinks go unnoticed, or nearly so, and the visual world is perceived as being continuous across blinks. Simulated blinks-interruptions of light with similar durations but with the eyes open-are perceived as much more salient than actual blinks [START_REF] Riggs | Suppression of the blackout due to blinks[END_REF]Volkmann et al., 1980). We use the term visual continuity to refer to this near-perfect suppression of the sensory consequences of blinks. Visual continuity in spite of frequent and otherwise salient blackouts is a fundamental problem for any theory of vision that takes the retinal input as a starting point.

In order to account for visual continuity, Volkmann et al. (1980) proposed that sensitivity to light was actively suppressed during blinks, similarly to saccadic suppression [START_REF] Volkmann | Human visual suppression[END_REF]Burr et al., 1994). Volkmann, Riggs and colleagues were able to measure visual thresholds during blinks by, ingenuously, lighting the retina from behind, by means of an optical fiber connected to a light source inside the mouth. They found a decrease in sensitivity of about 0.5 log units around blinks. This decrease started before the beginning of eyelid closing and ended around 300 ms after closure (Volkmann et al., 1980). The increase in threshold is caused by the closing rather than by the opening of the eyelid [START_REF] Volkmann | Measurements of visual suppression during opening, closing and blinking of the eyes[END_REF], increases as the amplitude of the eyelid motion goes from partial to full closure [START_REF] Stevenson | Dependence of visual suppression on the amplitudes of saccades and blinks[END_REF] and does not differ between reflexive and voluntary blinks [START_REF] Manning | Reflex eyeblinks and visual suppression[END_REF]. The interpretation given by Volkmann, Riggs and their colleagues was that the drop in visual sensitivity is caused by the corollary discharge of the blink, and that this drop accounts for visual continuity.

There is some physiological evidence in favor of an extra-retinal suppression of blinks. Gawne andMartin (2000, 2002) found that some neurons in areas V1-V4 in monkeys differentiate between visual interruptions due to blinks and those due to external darkening: the transient burst of activity present at the onset of an external darkening was absent at the onset of a blink, suggesting that there might be active suppression of this transient during blinks. [START_REF] Bristow | Blinking Suppresses the Neural Response to Unchanging Retinal Stimulation[END_REF] used the same paradigm as Volkmann et al. (1980), bypassing the eyelid, to compare BOLD activity during blinks to identical retinal input without blinks. They found no significant differences in LGN, V1 and V2 but a significant decrease in activity during blinks in higher areas: V3 and many regions of parietal and pre-frontal cortices. These results suggest that blink suppression occurs at later stages of visual processing.

Regardless of the origin of the small decrease in visual sensitivity around blinks, it seems unlikely that it could account for visual continuity: a 0.5 log unit decrease in sensitivity cannot mask 1.5-2.5 log unit drop in luminance. We therefore put forward a different hypothesis, that visual continuity around blinks is at least partly caused by an alteration in the perception of time and duration around blinks that makes their sensory consequences less salient. This hypothesis is motivated by an observation made by [START_REF] Riggs | Suppression of the blackout due to blinks[END_REF]: while simulated blinks (we'll call them blanks) seem much more salient than blinks of the same duration, much briefer blanks do appear similar to blinks. If the perceived duration of the visual interruption introduced by a blink is reduced, this might contribute to perceptual continuity across blinks. Closely related ideas about time perception and continuity across blinks have recently been put forward by [START_REF] Irwin | Perceiving a continuous visual world across voluntary eye blinks[END_REF] and [START_REF] Grossman | Where does time go when you blink?[END_REF]. The empirical results of these studies overlap with part of the results of our Experiment 2 (see below for further discussion).

The interruption caused by blinks has two consequences. The first is the period of darkness itself.

To what extent do we perceive it as briefer than its true duration, as predicted by our hypothesis?

In order to address this question, in Experiments 1a and 1b we artificially extended the period of darkness after the end of the blink, or before its beginning. In a task in which participants reported the combined duration of the blink-induced and artificial darkness, we used a multivariate analysis to evaluate the relative contribution of the two durations. We could therefore evaluate the perceived duration during the blink relative to blanks immediately preceding or following the blink. If the duration of blink-induced darkness is perceived differently than externally-induced darkness, we can conclude that the origin of the effect is extra-retinal.

The second consequence of blinks is the interruption of visual stimuli that straddle the blink. To what extent does the blink reduce the perceived overall duration of such stimuli? When participants are asked to judge the time interval between the initial onset and the final offset of a stimulus with a temporal gap, the perceived duration is smaller than for stimuli without gaps (e.g., [START_REF] Fortin | Timing during interruptions in timing[END_REF]. In Experiment 2, we checked whether this is true for stimuli interrupted by blinks as well as by blanks, and how the reduction in perceived duration depends on the duration of the blink or blank.

Experiment 1a: Perceived duration of blink compared to post-blink duration

Methods

Participants

Sixteen people took part in the experiment (mean age 23.5 years, range 20-28 years, 12 women).

All reported normal or corrected-to-normal vision. Except for the first author, all were naïve regarding the hypotheses and were paid 10€ per hour. All participants provided prior informed consent, in accordance with a procedure that adhered to the ethical principles of the Declaration of Helsinki. (The experiment was not reviewed by an ethics committee because in France legal ethics committees do not review noninvasive protocols.) Two of the 16 participants were excluded from the analysis for reasons given in the Results section below.

Sample sizes for all three experiments in this study were selected using pilot data for Experiment 2, from which we concluded that the relative main effect of blinks would be roughly 0.2: on the order of 100 ms for judgments of intervals of 500 ms. Assuming a relative standard deviation likewise of 0.2, and a power of 0.8, yields a sample size of 8. To be conservative, and in anticipation of participants unable to carry out the task, we fixed sample sizes in all experiments to be between 12 and 16 participants. Sample sizes were not changed after the experiment began.

Apparatus

The experiment took place in a dimly lit test room (0.02 cd/m 2 ). Visual stimuli were displayed using a PROPixx projector (VPixx Technologies Inc., Canada). This DLP projector was operated at an input rate of 120 Hz, and an output rate of 1440 Hz, at resolution 960 x 540 pixels in 8-bit grayscale. (Twelve frames were coded in each input frame, which the projector displays at 12 times its input frequency.) Moreover, the projector's three LED lamps can be controlled directly by the attached PC. Participants sat 190 cm from the projection screen that subtended 41 by 23 degrees of visual angle [dva]. The right eye was monitored using an EyeLink 1000 Plus desktop mount eyetracker (SR Research Ltd.) operated at 1000 Hz. A head and chin rest stabilized the head. The precise latencies of the projector and the eyetracker were determined using separate procedures, described in the Supplementary Materials.

Insert Figure 1 about here

Stimuli

There were two conditions, BLINK and REPLAY. The timeline of a BLINK trial is shown in Figure 1a. At the beginning of a trial, participants viewed a random texture of a low spatial frequency noise, generated by low-pass filtering random pixels (100 cd/m 2 ). After a random delay between 0.5 and 1 s, an auditory beep served as the signal for the participant to blink. The projector lamps, the only source of light in the experiment room, were turned off as soon as the beginning of the blink was detected (by which time the eyes were closed). As soon as the end of the blink was detected (which happened 58 ms after the actual end of the blink: see Supplementary Materials for details), the lamps were kept off for an additional duration drawn randomly from a uniform distribution between 0 and 200 ms. It was determined that the time required for the lamps to reach 50% luminance was 13 ms (Supplementary Materials). Therefore, blinks were followed by a period of darkness lasting between 71 and 271 ms. At the end of that period the projector lamps were turned back on and participants saw the random texture again. The reason for turning the lamps off during the blink was to make the darkness during the blink and the darkness after the blink as similar to each other as possible. As mentionned previously, light is attenuated incompletely by the eyelid during blinks, so the lamps had to be extinguished during the blink for the two types of 'darkness' to be roughly equally dark.

In the REPLAY condition (the data from which were only used to calibrate our data analysis procedure-see below), stimuli were identical to corresponding BLINK trials and presented in the same order, but participants were instructed not to blink. Projector lamps were turned off and back on at the same time as in corresponding BLINK trials.

Procedure

In the BLINK condition, participants were instructed to blink as naturally as possible as soon as they heard the auditory signal. To probe the perceived duration of darkness, we used the method of single stimuli [START_REF] Morgan | The use of an implicit standard for measuring discrimination thresholds[END_REF], asking participants to judge on each trial whether the total duration of darkness-caused by their blink and the additional period of darkness immediately following the blink-was longer or shorter than the average duration of total darkness experienced over the course of previous trials. Participants gave their response by pressing one of two keys. Trials where blink duration was above 250 ms, or where blink latency was below 100 ms or above 1 s, were identified online, discarded, and rerun later during the experiment.

In the REPLAY condition, participants were instructed not to blink, but the visual and auditory stimuli were identical, trial-by-trial, to BLINK trials in a preceding block. The task was also the same: to judge whether the period of darkness was briefer or longer than an internal standard.

The experiment began with a training block of 100 trials in the BLINK condition (not included in the analysis), followed by 4 experimental blocks of 300 trials each. In 15 out of 16 participants, a BLINK block was followed by a REPLAY block, followed by a second BLINK and a second REPLAY block. Because of operator error, in one participant the two BLINK blocks were performed first, followed by the two REPLAY blocks. Eleven participants ran the entire experiment in one session, while the remaining participants ran the experiment over two separate days.

Analysis

Determining accurate blink duration

Because knowing the actual duration of the blink is crucial to our experiment we wanted to check that the beginning and end of a blink, as detected by the eyetracker, corresponded to the actual complete occlusion of the pupil (our definition of a blink in this study). To do that we filmed the eyes of 3 additional participants on 100 trials with a 240 Hz camera (Hero 3, GoPro, California, USA) and compared the times of the beginning and end of blink, derived from manual frame-byframe by examination of the video, to those computed by the eye-tracker. We found a constant overestimation of the blink duration by the eyetracker and corrected for it in the subsequent analyses (see Supplementary Materials for details).

Offline trial selection

We removed the data of two participants from subsequent analysis because, in contrast to othre participants, their responses were flat as a function of duration (in other words, they were not performing the task).

We removed 3.5% (±3.3%: between-participant standard deviations will be noted using the "±" sign) of trials for which the duration of the blink was more than 3 standard deviations from the mean of each participant, or for which blink duration was null (i.e., no complete occlusion of the pupil).

Data analysis

We assumed that the perceived duration was a linear combination of the two subintervals, the blink and post-blink durations: 𝛽𝐷 𝐷 . We fitted the responses to a logistic function of this linear combination:

𝑅 ~ 1 exp 𝛼 𝛽𝐷 𝐷 𝐷 (1)
The three parameters of model (1) are:

 𝛽 : quantifies the relative contribution of post-blink darkness to overall duration.

Hypothetical data from the extreme cases of 𝛽 1 or 0 are shown in Figure 2a. If 𝛽 1, then blink and post-blink durations contribute equally to the perceived duration of darkness, which then would depend linearly on the sum of the two durations. This is the null hypothesis: durations within the blink are estimated no differently than durations outside of the blink. If, in contrast, 𝛽 0, then the duration of the blink does not contribute at all to the perceived duration of darkness, which would then depend only on the postblink duration. If 𝛽 is equal to 0.5 then a given duration is perceived as half as long during the blink as it is during the post-blink darkness.

 𝛼: quantifies the precision of overall timing performance. Higher values of slope 𝛼 imply lower variability when presented with the same stimulus, while the inverse of 𝛼 can be considered to be a measure of the just-noticeable difference.

 𝐷 : the point of subjective equality, the standard adopted by the participant as the "average" in the single-stimulus task

Insert Figure 2 about here

In most participants, mean blink durations change systematically over the course of the experimental session, and probably so does the implicit standard for the single-stimulus task. This makes it tricky to fit model (1) to the data of an entire session, because such a fit implicitly assumes that the parameters are stationary over time. We solved this problem by fitting the model in a sliding window of 50 trials. Further details concerning the fits can be found in the Supplementary Materials.

Results

Mean blink duration was 80 ms (±35 ms).

The raw data of four representative participants are shown in Figure 2b, along with the results of the model, giving the estimates of the 𝛽 parameter for each of the participants, and the corresponding line of subjective equality separating the brief and long durations. In each of the participants the fitted value of the 𝛽 parameter was below 1, meaning that in total duration judgments, the within-blink darkness was weighted less than post-blink darkness.

The distribution of individual 𝛽 coefficients for all participants is shown in the left panel of Figure 2c. The mean of the individual estimates was 𝛽 ̅ 0.57. This implies that, on average, the dark period during a blink is perceived as lasting a little over half as long as the dark period following the blink. The 95% confidence interval on 𝛽 ̅ was [0.46, 0.68], calculated using a bootstrap and resampling the individual point estimates 10 samples). This confidence interval excludes the null-hypothesis value 𝛽 1, at which durations are perceived in the same way during the blink and immediately following the blink. The result also excludes the value 𝛽 0, showing that the component of the total duration during the blink is not entirely ignored, either. The same objective duration of darkness during the blink is judged as being briefer than an equivalent duration after the blink, with the blink duration discounted by almost half.

Experiment 1b: Perceived duration of blink compared to pre-blink duration

The previous experiment showed that the blink contributes less than the post-blink period to the total perceived duration of darkness. This effect could be due to an underestimation of duration specifically during blinks. Instead, it could be due to the overestimation of the duration of the postblink darkness. Another possible explanation is the overestimation of duration at the start of an interval-imagine a stopwatch that starts too slow and accelerates to normal speed. To check whether these last two explanations could account for the effect found in Experiment 1a, we performed a similar experiment, but with the additional period of darkness preceding-rather than following-the blink.

Methods

Apparatus, stimuli, and analyses were almost identical to the previous experiment except that instead of remaining for an additional period after the blink, the dark period was presented before the blink.

Procedure

The timeline of the trials is shown in Figure 1b. On each trial, participants viewed a random texture filtered to include only low spatial frequencies (as in Experiment 1). They were instructed to blink as naturally as possible when they heard an auditory signal, and to try to maintain a regular interval between the beep and their blink. We used the median latency in previous trials of the same block to adjust the timing of the pre-blink dark period. The target duration of the preblink period was drawn from a uniform distribution between 0 and 200 ms; but because of eyetracker delays, the actual duration was slightly longer (see Supplementary Materials). As in Experiment 1, we kept the lamps off during the blink itself (in addition to the interval of darkness preceding the blink). After the beginning of the blink was detected online, we waited for the end of the blink, defined as 5 consecutive samples with measurable pupil size, before turning the projector lamps back on.

As previously, participants were asked to judge whether the total duration of darkness caused by their blink and the additional period before their blink on that trial was longer or shorter than the average duration of total darkness experienced over the course of previous trials. They gave their response by pressing one of two keys. Trials for which blink latency was below 100 ms or above 1 s were discarded and replayed later during the experiment. The experimental session included a training phase of 100 trials (not included in the analysis) and the main experiment (600 trials in 2 blocks of 300 trials separated by a break).

Analysis

Offline trial selection

We removed trials for which the duration of the blink or of the pre-blink was outside of 3 standard deviations from the mean of each participant. We additionally discarded trials with null blink duration (no complete pupil occlusion) and trials for which the online criteria for detecting the end of the blink failed (trials with a difference of more than 20 ms between our online detection of the blink and the detection of the end of the blink by the EyeLink). Because 2 participants had over 10% rejected trials, their data were excluded from further analyses. For the remaining participants, 2.9% (±1.4%) of their trials were discarded. In order to equalize as much as possible the distributions of additional darkness periods in Experiments 1a and 1b, we further discarded the 5% of the trials in which pre-blink darkness duration was 0 (i.e., the blink began before the projector lamps were turned off), or where it was greater than 271 ms (the maximum in Experiment 1a).

Statistics

We performed the same analyses as in Experiment 1a, except that, in model ( 1), the duration of pre-blink darkness, 𝐷 , replaced that of post-blink darkness. The data of 3 participants could not be fitted and they were excluded from further analyses. Thus, the results presented below are based on the data of 11 participants.

Results

Average blink duration was 118 ms (±41 ms) and the pre-blink darkness ranged between 1 ms and 271 ms, with an average duration of 111 ms (±7 ms).

We fitted the same model to the data as in Experiment 1a, with 𝐷 instead of 𝐷 . Individual estimates of 𝛽 are shown in Figure 2c. Averaging the coefficients of the individual participants, we find that mean 𝛽 ̅ 0.36, with a 95% confidence interval of [0.20, 0.53], calculated using the bootstrap. This means that, on average, the darkness during the blink was perceived as less than half as long as the darkness preceding the blink. The mean β coefficient was lower in Experiment 1b than Experiment 1a, but their difference was not significant, although it did approach significance (𝑡 1.98, two-tailed 𝑝 0.06).

Thus, the results of Experiments 1a and 1b taken together show that visual interruptions caused by blinks are systematically underestimated. Darkness caused by blinks is perceived to last about half as long as the darkness immediately preceding or following the blink.

Experiment 2 -Perceived duration of stimuli that straddle blinks

In Experiment 1, we found evidence that an extra-retinal signal around blinks reduces the perceived duration of blink darkness, compared to intervals immediately preceding or following the blink. Another effect of a blink is to 'puncture a hole' in stimuli that straddle the blink in time.

Does this reduced duration play a role in the perceptual suppression of the blink-induced hole?

To address this question we examined the perceived duration of stimuli that straddle blinks. First, we compared the perceived duration of visual stimuli punctured by blinks to unpunctured stimuli, occurring after the blink. Second, we compared the effect of blink-induced punctures to the effect of an identical but external optical puncture-a 'blank'-on perceived duration. If we interpret the results of Experiments 1a and 1b as signifying that the blink itself is perceived as being briefer than its true duration, we might expect that a stimulus straddling a blink be likewise perceived as briefer than an unpunctured stimulus.

Methods

Participants

Twelve people took part in the experiment. One participant who could not reach criterion on the pretest (see below) was excluded from the experiment. Data from the remaining 11 participants were included in the analysis (mean age: 23.5 years, range: 19-31, 3 men). None had previously participated in an experiment on perception around blinks and all were naïve regarding the hypotheses tested in the experiment. All reported normal or corrected-to-normal vision and no neurological disorders. They were paid 10€ per hour. The procedure adhered to the ethical principles of the Declaration of Helsinki.

Apparatus

The apparatus was the same as in Experiments 1a and 1b, except that participants sat at 130 cm from the projection screen that subtended 60 34 dva.

Insert Figure 3 about here

Design and stimuli

We used a within-participant design: every participant performed the two puncture conditions, BLINK and BLANK, and the two timing conditions, STRADDLE and LATER (Figure 3). Participants viewed a gray background (49 cd/m 2 ). On every trial a darker gray square (19 cd/m 2 , 5 dva side) was presented, whose duration was chosen from a uniform distribution between 250 and 500 ms.

The BLINK condition was always run first. Participants were instructed to blink upon hearing the go signal. In STRADDLE trials, the presentation of the square was timed so as to straddle the blink.

In LATER trials, the square was presented 200 ms after the end of the blink as detected online. In order for the blink to happen during the presentation of the square in the STRADDLE condition, we adjusted the timing of the onset of the square relative to the auditory go signal. To do that, we computed the median blink duration and latency since the beginning of the experimental block, and updated those values on every trial. Median blink duration and latency were combined with stimulus duration, with the goal of making the midpoint of the blink coincide with the midpoint of the stimulus. The BLANK condition was always run second, because the time course of the stimuli and order of trials was yoked to that of corresponding trials in the BLINK condition. The only difference was that the puncture was caused by turning all the pixels on the display to their lowest possible luminance (0.08 cd/m 2 ) for the same duration as the blink in a corresponding BLINK trial.

Procedure

All participants came to the laboratory on two separate days: they ran the BLINK condition on the first day and the BLANK condition on the second day.

Each experimental session consisted of 3 phases: a pretest, a training phase and a finally the main experiment. The task was the same for all phases: using the method of single stimuli [START_REF] Morgan | The use of an implicit standard for measuring discrimination thresholds[END_REF], we asked participants to press one of two keys to indicate whether the duration of the square was longer or briefer than its average duration. In the BLANK condition, participants were told that the screen would momentarily turn black but that they should not pay attention to this and focus on the duration of the stimulus. Participants were instructed to maintain fixation around screen center. (We considered explicitly instructing participants to report either the interval between the initial appearance and the final disappearance of the square stimulus, or, on the contrary, instructing them to discount any disappearance in their reported duration. However, we found that such instructions were impossible to follow, partly because the stimuli were so brief and partly because, at least in the BLINK condition, participants did not perceive the blink period as a disappearance. We therefore chose the most neutral instructions: evaluate whether the duration of the square was longer or briefer than the average.)

The goal of the pretest was to check that participants could perform duration discrimination. The 100 trials consisted of a simple presentation of the square stimulus lasting 250-500 ms, without any blinks or blanks, while participants performed the same duration task as in the main experiment, namely to classify the intervals as "brief" or "long." At the end of a pretest block we fitted responses to a logistic function of the duration of the square. If the resulting slope was above 0.01 ms the participant moved on to the training phase, and otherwise the pretest was run again. One participant who could not reach the criterion after three pretests was dropped from the experiment.

In the training and main phases that followed, participants heard a beep after a randomly chosen duration between 500 and 750 ms after the start of a trial. In the BLINK conditions participants were instructed to blink as naturally as possible as soon as they heard the beep. In the BLANK condition they were instructed not to blink, while fixating the center of the display. The training block consisted of 50 trials (25 each in STRADDLE and LATER) and the main phase consisted of 3 blocks of 120 trials (60 per timing condition). Only trials from the main experiment were used for the analyses. Each of the two sessions lasted between 1 and 1.5 hours.

Online trial criteria

To ensure a sufficient number of usable trials, we applied several online criteria in the BLINK condition. Only trials with blink duration below 250 ms were allowed. If duration was above this threshold, a sound informed the participant that their blink was too long (some participants at the beginning of the experiment made unnaturally long blinks). In the STRADDLE condition only trials for which the blink started after the beginning of the stimulus and ended before the end of the stimulus were considered valid. Trials that did not meet those criteria were discarded and replayed later on during the experiment.

Analysis

Offline trial selection

In the BLINK condition, we additionally removed trials for which the beginning of the blink occurred less than 30 ms after the onset of the square and trials for which the end of the blink occurred less than 30 ms before the offset of the stimulus. The corresponding trials in the BLANK conditions were also removed. Trials for which participants blinked before the go signal were also removed from further analysis. In the BLANK condition, we discarded trials for which participants blinked between the auditory stimulus and the end of the stimulus. We furthermore removed an additional 0.6% of the trials which had incorrect parameters due to a programming error discovered during the analysis. Finally, we removed any trials whose corresponding trial in the BLINK or BLANK condition was removed, ensuring that trials in the two timing conditions remained matched.

Models

As in Experiments 1a and 1b, we assumed that the perceived duration is a linear combination of the total stimulus duration 𝐷 (i.e., the time from initial stimulus onset to its final offset, 250-500 ms), and the duration of the blink or blank, 𝐷 / . We fit responses to a model similar to that for Experiments 1a and 1b:

𝑅 ~ 1 exp 𝛼 𝐷 𝛽𝐷 / 𝐷 (2)
The three parameters of model ( 2) are:

 𝛽: quantifies the fraction of blink or blank duration is subtracted from overall perceived duration. If 𝛽 0 then the blink or blank does not affect the perception of stimulus duration, which would then depend only on the time interval between initial stimulus appearance and its final disappearance. If, on the contrary, 𝛽 1, then the perceived stimulus duration depends on total stimulus duration minus the duration of the puncture-in other words, the duration that the stimulus is visible. If 𝛽 is equal to 0.5 then perceived duration depends on a variable equal to the total duration minus half the duration of the puncture.

 𝛼: quantifies the precision of overall timing performance, as in model ( 1) for Experiments 1a and b. Higher values of slope 𝛼 imply lower variability when presented with the same stimulus, while the inverse of 𝛼 can be considered to be a measure of the just-noticeable difference.

 𝐷 : the point of subjective equality, the standard adopted by the participant as the "average" in the single-stimulus task, as in model ( 1) for Experiments 1a and b

The fit of model (2) was performed separately for each of the four combinations of the puncture conditions (BLINK, BLANK) and timing conditions (STRADDLE, LATER). The analysis was performed in the same way as for Experiments 1a and 1b, using a sliding window of trials (see

Supplementary Materials for details).

A separate, simplified 2-parameter model was also fitted to the data using the same procedures as model ( 2):

𝑅 ~ 1 exp 𝛼 𝐷 𝐷 (3)

Results

Participants ran on average 1.2 pretest blocks in the BLINK condition and had a mean slope of 0.024 (±0.0079) on their final pretest block. In the BLANK condition, they ran a mean of 1.1 pretest sessions and had a mean slope of 0.023 (± 0.0088). 7.1% (5.2%) of the trials in the main experimental blocks were excluded according to the criteria listed above. Mean blink duration was 81 ms (±36 ms).

Insert Figure 4 about here Examples of responses as a function of stimulus duration, in the four conditions, are shown in Figure 4 for three representative participants. The fraction of "long" responses increases with stimulus duration with a sigmoid shape, showing that participants successfully performed the duration judgment task. An effect of the discrete conditions that can be seen in Figure 4 (and as will be borne out by the analysis below) is that the STRADDLE curves lie below the LATER curves.

This means that a stimulus, when punctured by a blink or blank, is perceived as briefer than a stimulus of the same duration when unpunctured. In other words, the point of subjective equality is higher for STRADDLE than for LATER. However, there does not seem to be a systematic difference between BLINK and BLANK conditions, at least as their dependence on stimulus duration is concerned-again, this will be borne out by the analyses below.

Insert Figure 5 about here If punctured stimuli are perceived as briefer, does this decrease in perceived duration depend on the duration of puncture? To answer this question, we examined the effect on responses of the second continuous variable, blink or blank duration, which varied from trial to trial. Figure 5 illustrates mean responses in the STRADDLE condition, as a function of both stimulus duration and blink/blank duration. The response patterns in the BLINK and BLANK conditions are quite different. In both conditions, we see a strong dependence of reported duration on stimulus duration (as already seen from the rising curves in Fig. 4)-the contours are mainly vertical.

However, the two conditions differ in their dependence on blink or blank duration. In the BLINK condition, on the one hand, there is little systematic dependence of reported durations on the blink duration. In the BLANK condition, on the other hand, the sloping contours indicate a systematic dependence of perceived duration on the duration of the blank: the longer the blank, the briefer the perceived stimulus. Thus, the decrease in perceived duration due to the puncture seems to depend on the duration of the puncture in the BLANK, but not in the BLINK, condition.

Insert Figure 6 about here

We therefore decided to fit the responses to the 3-parameter model ( 2), with the perceived duration a linear combination of stimulus and blink/blank durations, separately for each of the four conditions. The results for the 𝐷 and 𝛽 parameters of the model are shown in Figure 6. An ANOVA on 𝐷 showed only a main effect of timing condition: 𝐷 was about 76 ms higher in the STRADDLE than in the LATER conditions (Figure 6a; 𝐹 , 57.7, 𝑝 0.001 . This means that, all other factors being equal (durations of the stimulus and of the blink or blank), the punctured STRADDLE stimuli are perceived as being 76 ms briefer on the average than the unpunctured LATER stimuli. This difference is quite close to the mean duration of the blink or blank, 81 ms.

Thus, punctured stimuli are judged to have a duration roughly equal to the total duration minus the duration of the puncture. However, the amount of time subtracted due to the puncture seems to be no different for blinks than for blanks, because there was no significant interaction between timing and puncture conditions (𝐹 , 1.86, 𝑝 0.1 .

However, the behavior of the 𝛽 (Fig. 6b) parameter shows that there is, in fact, a difference between BLINK and BLANK. In both LATER conditions the mean values of 𝛽 are close to zero (𝛽 ̅ 0.09 0.20, 0.45 for BLINK, 0.08 0.31, 0.15 for BLANK, with bootstrap-derived 95% confidence intervals given in square brackets); both confidence intervals include zero. This is not surprising: because the blink or blank falls outside the stimulus, it makes sense that its exact duration should have no effect on the perceived duration of the stimulus. The same is true for blinks in the STRADDLE condition: 𝛽 ̅ 0.28 0.09, 0.65 . However, this is not true for blanks in STRADDLE, where we find that 𝛽 ̅ 0.84 0.58, 1.10 , and therefore significantly greater than zero. A paired t test on individual estimates revealed that 𝛽 was significantly greater for blanks than for blinks ( 𝑡 3.43, 𝑝 0.006 ). Similarly, a repeated-measures ANOVA revealed a significant interaction between puncture and timing conditions (𝐹 , 14.4, 𝑝 0.003).

Finally, the results for 𝛼, the slope of the psychometric curve, are shown in Fig. 6c. Contrary to what one might expect, the slopes are larger in the STRADDLE condition, where the stimulus is punctured, than in the LATER condition, where it is not (𝛼 21.5 in STRADDLE and 19.2 in LATER). This difference is confirmed by an ANOVA, where the main effect of timing condition is the only significant effect (𝐹 , 8.64, 𝑝 0.015).

Thus, the perceived duration of a stimulus that straddles a blink either does not take into account the actual duration of the blink, in a trial-to-trial way, or does so significantly less than when stimuli straddle blanks. Does this mean that stimuli that straddle blinks are perceived as longer than those that straddle blanks? To find out, we fitted the data to the simplified model (3), which only takes into account stimulus duration (but not blink/blank duration). In this model, variations in the 𝐷 parameter quantify differences in perceived duration. We found effects very similar to the ones shown in Fig. 6a, namely large values of 𝐷 in STRADDLE (453 [429,474] for BLINK,468 [440,490] for BLANK) and smaller ones for LATER (346 [328, 364] for BLINK,357 [336,376] for BLANK). A repeated-measures ANOVA shows only a main effect of timing condition (𝐹 , 78.7, 𝑝 0.0001).

Thus, intervals punctured by blinks seem briefer than unpunctured ones. The average reduction in perceived duration is 98 ms (full model ( 2)), calculated by taking the difference of the values of 𝐷 between the STRADDLE and LATER conditions. We have shown that, in contrast to blanks, this value is not correlated from trial to trial with the actual duration of each blink-in this sense we can say that the value of the duration reduction is generic. Where can this generic value come from? As we mentioned above, the fact that the reduction in perceived duration is close to the average blink duration-81 ms in this experiment-strongly suggests that the two durations are related. A signal of this relationship would be a within-participant correlation between duration reductions and blink durations, in this condition. We found that the Pearson correlation coefficient between them was 𝑅 0.27, but a bootstrap test showed that the correlation was not significantly different from zero (𝑝 0.26). Therefore we cannot exclude such a correlation, but our experiment does not have enough statistical power to reveal it.

To summarize the results of this experiment: a stimulus punctured by either a blink or a blank is perceived as briefer than an unpunctured stimulus. In the case of blanks, this reduction reflects the actual duration of the blank on each trial. For blinks, however, the reduction in duration is independent of the blink duration, and seems to be a generic value close to the average blink duration.

Discussion

In two experiments we have shown that the perceived duration of darkness during blinks differs from externally caused darkness. The first experiment directly compared blink-induced to externally produced darkness, either directly preceding or directly following a blink. We found that about half of the duration of the blink-induced darkness was discounted with respect to external darkness. In the second experiment we found that the duration of visual stimuli optically punctured by blinks or blanks was perceived as briefer than that of unpunctured stimuli, regardless of whether the puncture was caused by a blink or a blank (i.e., an externally produced period of darkness). However, in the case of blanks the reduction in perceived duration accurately reflected the duration of the blank, which varied from trial to trial; whereas in the case of blink, the reduction depended much less or not at all on actual blink duration.

The results of Experiments 1a and 1b show that an extra-retinal signal is involved in modulating time perception around blinks, because there is no retinal difference between pre-, intra-, and post-blink darkness. Likewise for Experiment 2: even though the blank condition simulated the retinal information during blinks, the dependence on blink or blank duration differed in the two conditions. Previous studies have found an effect of ocular saccades on temporal perception [START_REF] Morrone | Saccadic eye movements cause compression of time as well as space[END_REF][START_REF] Yarrow | Illusory perceptions of space and time preserve cross-saccadic perceptual continuity[END_REF], but more recent work has shown that those effects can be replicated using optically simulated saccades [START_REF] Knöll | Spatio-temporal topography of saccadic overestimation of time[END_REF][START_REF] Zimmermann | Masking produces compression of space and time in the absence of eye movements[END_REF]. In contrast, the results that we report here, a reduced perceived duration of blink-induced darkness, do seem to rely on extra-retinal signals.

One of the two results of Experiment 2 was the same for blinks and blanks: stimuli punctured by both blinks and blanks appeared briefer than stimuli without the puncture by about 100 ms, the approximate mean duration of the blinks and blanks. It can also be compared to past studies of slightly longer stimuli, between 0.5 and several seconds, that also found that punctured stimuli are perceived as briefer than unpunctured ones [START_REF] Fortin | Timing during interruptions in timing[END_REF][START_REF] Fortin | Expectancy in humans in multisecond peak-interval timing with gaps[END_REF]. Interestingly, the presence of the puncture did not decrease the slopes of the psychometric curves, which indicates that the subtraction was probably perceptual rather than deliberate or decisional. It is also notable that, although in stimuli punctured by blanks the interruption appeared much more salient and longer-lasting than in stimuli punctured by blinks, their reported durations were about equal. However, another result of Experiment 2 did distinguish between blinks and blanks:

for blanks the amount by which the stimulus appeared briefer tracked trial-by-trial variations in the blank durations. In contrast, for blinks the reduction in perceived duration was much less correlated to actual blink duration on each trial, seeming to be a default value-close to the duration of a typical blink. While we cannot pinpoint the precise mechanism for this effect, there is an argument for why it makes sense. Although the duration of blinks varies from one blink to the next, the variation is not very large. In Experiment 2, for instance, mean duration was 81 ms, and its mean within-participant standard deviation was 25 ms. Thus, the visual system can assume a certain blink duration instead of measuring it without introducing too much error (after all, with slopes of about 20 ms , JNDs are roughly 50 ms-above the typical variability for blink durations). Blanks, on the other hand, can have any duration whatsoever, and thus it is important to measure their duration.

Another difference between blinks and blanks in Experiment 2 is the voluntary nature of the blinks in contrast to the involuntary nature of blanks. This difference may have a role to play in explaining the different effects that we found of the two types of interruption on the perception of duration, given connections that have been established between voluntary action and time perception [START_REF] Haggard | Voluntary action and conscious awareness[END_REF]. It would therefore be interesting in future studies to vary the voluntary/involuntary and blink/blank dimensions in more systemic ways.

There is a seeming contradiction between the results of the first and second experiments. In Experiments 1a and 1b we showed that the durations of blinks are underestimated by about 50%

as compared to the intervals of darkness or blanks immediately preceding or following the blink.

In the second experiment we found that punctured intervals are perceived as briefer than unpunctured ones-but that this reduction in perceived duration is not significantly smaller for blinks than for blanks. This would be a genuine contradiction if we assumed a single timing mechanism for all visual stimuli. In recent years, however, the notion that there exist multiple timing mechanisms, even within single modalities, has become widespread [START_REF] Bruno | Multiple channels of visual time perception[END_REF][START_REF] Gorea | Ticks per thought or thoughts per tick? A selective review of time perception with hints on future research[END_REF]. Thus, we have some evidence that the timing of blink-induced darkness, and of brief stimuli punctured by blinks, may be performed by distinct timing mechanisms. From a functional perspective it might be useful for the visual system to attenuate the perceived duration of the blackout caused by blinks, for the same reason that it is useful to attenuate the perceptual consequences of actions in general. However, when estimating the duration of external stimuli interrupted by blinks, a generic time interval is discarded from the duration of the stimulus, likely due to the fact that the external world usually does not change much in the blink of an eye, making it unnecessarily costly to keep precise track of the duration of the interruption.

It is also worth pointing out that Experiments 1 and 2 may not actually be in contradiction to one another, simply because they test quite different situations. In Experiment 1 the objective total duration is 𝐷 𝐷 / , so if the observer were veridically reporting the total duration, we would find that 𝛽, the coefficient of 𝐷 in our model 𝛽𝐷 𝐷 / , to be equal to 1. In contrast, in Experiment 2 there is no clear objective total duration. On one hand, one could fill in the interruption, in a temporal analogue to the spatial filling-in of scotomas. In that case, the correct duration would be 𝐷 , and the coefficient 𝛽 in our model 𝐷 𝛽𝐷 would be equal to 0. On the other hand, the dark interval could be subtracted from the perceived duration, in which case we would have 𝛽 1. The point is that the status of the 𝛽 parameter is quite different in Experiments 1 and 2, despite the similar-looking models. As it turns out, there is a decrease in perceived duration due to the interruption, but that the decrease depends differently on the duration of blinks than on the duration of blanks. In short, the results of Experiments 1 and 2 may not be immediately comparable because of the differences in the stimuli.

Two recent studies have put forward a hypothesis compatible with ours, namely that alterations in time perception help to bring about visual continuity during blinks [START_REF] Grossman | Where does time go when you blink?[END_REF][START_REF] Irwin | Perceiving a continuous visual world across voluntary eye blinks[END_REF]. Both of these studies feature experiments similar to a subset of our Experiment 2, in which the perceived duration of visual intervals punctured by blinks is compared to unpunctured intervals-and both find results similar to our Experiment 2, namely that blinkpunctured intervals are perceived as briefer than unpunctured intervals, by about the average duration of the blink. It is interesting to note that one of these studies was carried out using spontaneous blinks [START_REF] Grossman | Where does time go when you blink?[END_REF] rather than the voluntary blinks used here. As mainly automatic actions, these spontaneous blinks are at least partially physiologically distinct from the voluntary blinks in our study [START_REF] Vanderwerf | Eyelid Movements: Behavioral Studies of Blinking in Humans Under Different Stimulus Conditions[END_REF]. Nevertheless, the fact that perceived interval duration is discounted by the same amount, on the average, for involuntary as for voluntary blinks, and by a similar amount for completely involunary blanks, shows once again that the decrease in perceived duration, at least on the average, may be independent of the detailed motor aspects of the interruption.

In addition to the discounting of the perceived duration of blink-punctured intervals that we find in common with the two recent studies [START_REF] Grossman | Where does time go when you blink?[END_REF][START_REF] Irwin | Perceiving a continuous visual world across voluntary eye blinks[END_REF], here we also compared intervals punctured by blinks to those punctured by blanks, and found that both are perceived as briefer, by about the same amount. Furthermore, we were able to demonstrate a difference between blinks and blanks: whereas, for blanks, the shortening of perceived duration depends on the actual duration of the blanks, this is not the case for blinks.

The discounted perception of blink durations (Experiment 1) or their unavailability for further processing (Experiment 2) could be a special case in which the reafferent sensory effects of our motor actions are reduced or discarded from perceptual processing (see, for example, [START_REF] Blakemore | Central cancellation of self-produced tickle sensation[END_REF][START_REF] Cardoso-Leite | A new look at sensory attenuation. Action-effect anticipation affects sensitivity, not response bias[END_REF]. A corollary signal could achieve this discounting by dampening general processing of that informationaccounting for diminution of brain activity found around blinks (Bristow, Frith, et al., 2005;[START_REF] Bristow | Blinking Suppresses the Neural Response to Unchanging Retinal Stimulation[END_REF] and inhibition of the neural reaction to the onset of the interruption of light [START_REF] Gawne | Activity of Primate V1 Cortical Neurons During Blinks[END_REF], 2002). Time compression could be a behavioral marker of this decreased processing, in agreement with the notion that the amount of energy committed to processing environmental stimuli accounts for the perception of duration [START_REF] Eagleman | Is subjective duration a signature of coding efficiency?[END_REF]see Wittmann & van Wassenhove, 2009 for a theoretical review), or, more The fact that the estimates of 𝛽 are significantly smaller than 1 implies that the duration of the blink is estimated with a lower gain than the periods of darkness immediately preceding or following the blink. Here, only the main effect of timing condition is significant, with 𝐷 in the STRADDLE conditions significanly higher than in the LATER conditions. This means that a stimulus punctured by either a blink or a blank is perceived as significantly briefer than an unpunctured stimulus. (b) The 𝛽 parameter, quantifying how much of the interruption duration is subtracted from the total duration. In the STRADDLE condition, 𝛽 is significantly higher in the BLANK than in the BLINK condition (it is not significantly different from 0 in BLINK, and not significantly different from 1 in BLANK). It is also not significantly different from 0 in the LATER conditions. Thus, when a visual stimulus is punctured, participants seem to have less access to the duration of the puncture for blinks than for blanks. (c) The 𝛼 parameter, which is the slope of the psychometric curve. The only significant effect is that of the timing condition, showing that the slope is significantly greater in the STRADDLE condition where the stimulus is punctured by a blink or a blank than in the LATER condition where it is not punctured.

Participants

  16 participants ran the experiment (mean age: 24.7 years, range: 20-30 years, 6 men). All reported normal or corrected-to-normal vision and were naïve regarding the hypotheses tested in the experiment. None had participated in the previous experiment. They gave informed consent and were paid 10€ per hour. The procedure adhered to the ethical principles of the Declaration of Helsinki.

FiguresFigure 1 .

 1 Figures

  Figure 1. (a) The timeline of a trial in the BLINK condition in Experiment 1a. Here, darkness is maintained for a variable duration following a blink. The blink latency and duration are given as mean values with between-participant standard deviations. The durations of the delay preceding the go signal, and the post-blink darkness were drawn randomly from uniform distributions with the limits shown. (b) The timeline of a trial in Experiment 1b, where the duration of the blinkinduced darkness is artificially lengthened by a preceding period of darkness.

Figure 2 .

 2 Figure 2. Results of Experiments 1a and 1b and their interpretation. (a) In our presentation of the raw data, each trial is shown as a dot, with the horizontal coordinate representing blink duration, and the vertical coordinate representing the duration of the additional darkness period, either post-blink (Experiment 1a) or pre-blink (Experiment 1b). The dot's color codes the response: blue for "brief" and green for "long". The two graphs illustrate hypothetical data at the two limits of our model, in which perceived duration is proportional to a linear combination of blink and post/pre-blink durations, 𝛽𝐷 𝐷 . The top graph shows hypothetical data for 𝛽 1, when the perceived duration depends veridically on the total duration, 𝐷 𝐷 . The bottom graph shows the case 𝛽 0 , when perceived duration depends on 𝐷 alone (and therefore the blink duration is discounted completely). (b) Data of four representative participants in Experiment 1a. The lines of subjective equality as determined by the model are shown in red, and corresponding estimates of 𝛽 are given above each graph. (c) The estimated values of 𝛽 for the individual participants in Experiment 1a ("POST") and Experiment 1b ("PRE") are shown in gray. The four participants whose data are shown in panel (b) are highlighted with dashed lines. The estimated population mean values of 𝛽 are shown as dots for the two experiments, with the bars showing the bootstrap-derived 95% confidence intervals of the mean.

Figure 3 .

 3 Figure 3. Time course of trials in Experiment 2. In the STRADDLE condition, the stimulus straddles the interruption (blink or blank), whereas in the LATER condition, the stimulus is presented 200 ms after the blink or the blank. When we talk about "stimulus duration," we mean the time from the initial appearance of the stimulus to its final disappearance.

Figure 4 .

 4 Figure 4. Data of Experiment 2, for 3 representative subjects (the three columns). The curves show smoothed responses (expressed as the fraction of "long" responses) as a function of total stimulus duration (scale shown on the bottom right graph). The red curves show the STRADDLE condition, in which the visual stimulus straddles the blink or blank, while the blue-gray curves show the LATER condition, in which the stimulus occurs after the blink or blank. The top panels show the BLINK condition, while the bottom panels show the BLANK condition.

Figure 6 .

 6 Figure 6. Model results of Experiment 2. (a) The constant 𝐷 parameter in the three-parameter model. The four combinations of interruption and timing conditions are shown in separate columns. Gray horizontal lines show parameter estimates for individual participants, dots sample means, and vertical bars bootstrap-derived 95% confidence intervals of the means. Here, only the

Acknowledgments

This work was supported by a doctoral fellowship given to MD by the Direction Générale de l'Armement. We thank Matteo Lisi for his generous help with the data analysis, and Arielle Veenemans for help in testing.

generally, with distributed mechanisms of time perception, in which perceived duration of perceptual events is derived directly from the operation of intrinsic perceptual circuits [START_REF] Ivry | Dedicated and intrinsic models of time perception[END_REF][START_REF] Muller | Perceiving the passage of time: Neural possibilities[END_REF].

A spatial analog of the temporal interruption of visual information by blinks is the blind spot, which is perceptually filled in. Two types of theories have been proposed for the filling-in [START_REF] Weil | A new taxonomy for perceptual filling-in[END_REF]: either the visual system simply ignores the absence of information, or it actively fills in the missing area of the visual field. These theories can be translated to the temporal gap created by blinks: does the visual system ignore the gap, in effect 'going to sleep' at the start of the blink and 'waking up' near the end, or does it actively interpolate information across the gap? Our results, showing that information about durations being discounted or becoming unavailable across the blink suggest that the gap is simply ignored.

Author contributions

All authors contributed to the study design. Stimuli were programmed by M. Duyck and M. Wexler. Data was collected by M. Duyck. M. Duyck and M. Wexler performed the data analysis. All authors contributed to the manuscript and approved the final version for submission. Responses are coded in shades of gray, with lighter shades indicating more frequent "long" responses. In the BLINK condition the contour lines are mainly vertical, which means that there is little systematic dependence on blink duration. In the BLANK condition, in contrast, the contours are diagonal, especially on the right-hand side of the graph. The diagonal contours indicate that stimuli were systematically reported as briefer with increasing blank duration.

(Individual data were smoothed using a Gaussian kernel with width 20 ms. Data for all participants have been averaged by normalizing each participant's blink/blank duration range and averaging the normalized data.)


Rinske M Alkemade

Marjolein De Jager

Berend Van Der Meer

Frank Smallenburg

Laura Filion

Point Defects in Crystals of Charged Colloids Point Defects in Crystals of Charged Colloids

Charged colloidal particles -both on the nano and micron scales -have been instrumental in enhancing our understanding of both atomic and colloidal crystals. These systems can be straightforwardly realized in the lab, and tuned to self-assemble into body-centered cubic (BCC) and face-centered cubic (FCC) crystals. While these crystals will always exhibit a finite number of point defects, including vacancies and interstitials -which can dramatically impact their material properties -their existence is usually ignored in scientific studies. Here, we use computer simulations and free-energy calculations to characterize vacancies and interstitials in both FCC and BCC crystals of point-Yukawa particles. We show that, in the BCC phase, defects are surprisingly more common than in the FCC phase, and the interstitials manifest as so-called crowdions: an exotic one-dimensional defect proposed to exist in atomic BCC crystals. Our results open the door to directly observing these elusive defects in the lab.

I. INTRODUCTION

Suspensions of charged colloids are among the most fundamental systems in colloidal science. These systems, consisting of charged colloidal spheres suspended in a solvent containing salt, which screens the Coulombic repulsions between the spheres, have been extensively studied using experiments, simulations, and theory [START_REF] Alexander | Charge renormalization, osmotic pressure, and bulk modulus of colloidal crystals: Theory[END_REF][START_REF] Kremer | Phase diagram of yukawa systems: model for charge-stabilized colloids[END_REF][START_REF] Robbins | Phase diagram and dynamics of yukawa systems[END_REF][START_REF] Monovoukas | The experimental phase diagram of charged colloidal suspensions[END_REF][START_REF] Sirota | Complete phase diagram of a charged colloidal system: A synchro-tron x-ray scattering study[END_REF][START_REF] Hamaguchi | Triple point of Yukawa systems[END_REF][START_REF] Hynninen | Phase diagram of hard-core repulsive Yukawa particles with a density-dependent truncation: a simple model for charged colloids[END_REF][START_REF] Yethiraj | A colloidal model system with an interaction tunable from hard sphere to soft and dipolar[END_REF][START_REF] Hsu | Charge stabilization in nonpolar solvents[END_REF][START_REF] Royall | Re-entrant melting and freezing in a model system of charged colloids[END_REF][START_REF] Masri | Measuring colloidal forces from particle position deviations inside an optical trap[END_REF][START_REF] Smallenburg | Phase diagrams of colloidal spheres with a constant zeta-potential[END_REF][START_REF] Kanai | Crystallization and reentrant melting of charged colloids in nonpolar solvents[END_REF][START_REF] Arai | Surface-assisted single-crystal formation of charged colloids[END_REF][START_REF] Chaudhuri | Triple junction at the triple point resolved on the individual particle level[END_REF] . In the case of singlecomponent, spherical colloids, the bulk phase behavior is extremely well understood, with impressive quantitative comparisons between theory and experiment [START_REF] Monovoukas | The experimental phase diagram of charged colloidal suspensions[END_REF][START_REF] Sirota | Complete phase diagram of a charged colloidal system: A synchro-tron x-ray scattering study[END_REF][START_REF] Kanai | Crystallization and reentrant melting of charged colloids in nonpolar solvents[END_REF] . These quantitative comparisons have been facilitated by the highly tunable nature of experimental systems of charged colloids [START_REF] Yethiraj | A colloidal model system with an interaction tunable from hard sphere to soft and dipolar[END_REF][START_REF] Van Gruijthuijsen | Sterically stabilized colloids with tunable repulsions[END_REF][START_REF] Kodger | Precise colloids with tunable interactions for confocal microscopy[END_REF] . From these studies we know that for sufficiently high densities or strongly charged particles, identically charged colloids self-assemble into one of two crystal structures, depending on the degree of screening. Broadly, for low salt concentrations, where the screening is weak, the system forms a bodycentered-cubic (BCC) crystal, while high salt concentrations result in a face-centered-cubic (FCC) crystal.

In equilibrium, such crystalline phases always feature a finite concentration of defects. These defects, like vacancies and interstitials, can have a profound impact on the mechanical, optical, and electronic properties of crystalline materials. In the realm of colloid science, where the creation of new materials to manipulate light is one of the overarching goals, the presence of defects strongly affects optical properties [START_REF] Yan | Incorporation of point defects into self-assembled three-dimensional colloidal crystals[END_REF][START_REF] Rengarajan | Effect of disorder on the optical properties of colloidal crystals[END_REF][START_REF] Nelson | Epitaxial growth of three-dimensionally architectured optoelectronic devices[END_REF] . It is therefore perhaps surprising, that despite the massive body of literature on crystals formed by charged colloids, little is known about how defects manifest in their 3d crystalline phases.

In three dimensions, some of the earliest work on defects in colloidal crystals focused on point defects (vacancies and interstitials) in single-component hard-sphere crystals [START_REF] Bennett | Studies in molecular dynamics. IX. Vacancies in hard sphere crystals[END_REF][START_REF] Pronk | Point defects in hard-sphere crystals[END_REF][START_REF] Pronk | Large effect of polydispersity on defect concentrations in colloidal crystals[END_REF] . This colloidal model system forms an FCC crystal, with relatively few point defects in equilibrium: at melting the crystal a) l.c.filion@uu.nl is predicted to have approximately 10 -4 vacancies and 10 -8 interstitials per lattice site. Subsequent studies have explored e.g. the local structural impact of defects [START_REF] Lin | Measuring nonlinear stresses generated by defects in 3d colloidal crystals[END_REF][START_REF] Van Der Meer | Diffusion and interactions of point defects in hard-sphere crystals[END_REF][START_REF] Vansaders | Strain fields in repulsive colloidal crystals[END_REF] , the diffusion of vacancies and interstitials [START_REF] Van Der Meer | Diffusion and interactions of point defects in hard-sphere crystals[END_REF][START_REF] Bennett | Persistence of vacancy motion in hard sphere crystals[END_REF] , and the emergence of stacking faults [START_REF] Hoogenboom | Stacking faults in colloidal crystals grown by sedimentation[END_REF][START_REF] Pronk | Can stacking faults in hard-sphere crystals anneal out spontaneously?[END_REF][START_REF] Marechal | Stacking in sediments of colloidal hard spheres[END_REF][START_REF] Pusey | Structure of crystals of hard colloidal spheres[END_REF] in hard-sphere systems. However, in general, defects in 3d colloidal crystals have received relatively little attention due, at least in part, to the expectation that they do not occur in large quantities in equilibrium.

A notable exception is the relatively recent prediction that simple cubic crystals of repulsive particles frequently exhibit large numbers (≈ 0.06 per lattice site) of vacancies that are spread over a row of lattice sites in one dimension [START_REF] Van Damme | Phase and vacancy behaviour of hard "slanted" cubes[END_REF][START_REF] Van Der Meer | Revealing a vacancy analog of the crowdion interstitial in simple cubic crystals[END_REF][START_REF] Smallenburg | Vacancystabilized crystalline order in hard cubes[END_REF] . These 1d vacancies, predicted for simple cubic crystals, are reminiscent of so-called interstitial crowdions. This intriguing type of interstitial defect was proposed by Paneth in 1950 [START_REF] Paneth | The mechanism of self-diffusion in alkali metals[END_REF] , to explain anomalous self-diffusion in BCC crystals of alkali metals. In this picture, the defect is expected to spread out over multiple lattice sites arranged along a one-dimensional line, resulting in preferential diffusion along that direction. In the atomic realm, explorations of these defects has been largely focused on simulations [START_REF] Derlet | Multiscale modeling of crowdion and vacancy defects in body-centered-cubic transition metals[END_REF][START_REF] Nguyen-Manh | Self-interstitial atom defects in bcc transition metals: Group-specific trends[END_REF][START_REF] Osetsky | Onedimensional atomic transport by clusters of self-interstitial atoms in iron and copper[END_REF][START_REF] Han | Self-interstitials in V and Mo[END_REF][START_REF] Zepeda-Ruiz | Strongly non-arrhenius self-interstitial diffusion in vanadium[END_REF] , and simple theoretical models used to capture their behaviour [START_REF] Kontorova | On the theory of plastic deformation and twinning[END_REF][START_REF] Landau | Model of interacting atomic chains and its application to the description of the crowdion in an anisotropic crystal[END_REF][START_REF] Kovalev | Theoretical description of the crowdion in an anisotropic crystal based on the Frenkel-Kontorova model including and elastic three-dimensional medium[END_REF][START_REF] Braun | Nonlinear dynamics of the Frenkel-Kontorova model[END_REF][START_REF] Dudarev | Coherent motion of interstitial defects in a crystalline material[END_REF][START_REF] Fitzgerald | Peierls potential for crowdions in the bcc transition metals[END_REF] .

To date, an analogue to these defects in a colloidal realization of a BCC crystal -which would allow for direct observation in real time using e.g. confocal microscopy -is lacking. It is therefore intriguing to explore how interstitials manifest in colloidal BCC crystals, and in particular in systems that can be directly, and even quantitatively, reproduced in an experimental setting.

Here we use computer simulations to explore both the concentration and structure of point defects in both BCC and FCC crystals of one of the most fundamental models for screened charged particles -the point Yukawa model. Our results predict that this fundamental system forms a direct colloidal realization of crowdion interstitials in BCC crystals. Moreover, we find that BCC exhibits significantly higher concentrations of point defects, and hence expect that these crowdions play an important role in controlling the material properties of the crystal. Importantly, given the substantial concentration of crowdions predicted to occur in equilibrium, our results pave the way to directly observing these rare and elusive defects in colloidal experiments.

II. MODEL

We consider a system of N charged colloids of diameter σ suspended in a solvent containing ions characterized by an inverse Debye screening length κ D and Bjerrum length λ B . Within Derjaguin-Landau-Verwey-Overbeek (DLVO) theory, the effective potential between the colloids is given by

β φ (r) = ε r e -κ D r , (1)
where

ε = Z 2 λ B e κ D σ (1 + κσ /2) 2 (2)
with Z the charge of the colloids in electron charges, and β = 1/k B T , with k B the Boltzmann constant and T the temperature. Note that this so-called Yukawa potential describes, not only charged colloids, but also has been widely applied in the study of dusty plasmas [START_REF] Ivlev | Complex plasmas and colloidal dispersions: particle-resolved studies of classical liquids and solids[END_REF] . Conveniently, the phase behaviour of this system can be fully characterized by two dimensionless parameters, namely

Γ = ε ak B T , κ = aκ D , with a = 4πN 3V - 1
3 the Wigner Seitz radius. The phase diagram for this system has been explored extensively using theory, simulations, and experiments. In Fig. 1, we show the phase behavior, using the phase boundaries approximated in Refs. 6 and 7. It consists of a fluid phase and two crystal phases: face-centered cubic (FCC) and bodycentered cubic (BCC), with all phase boundaries corresponding to first-order phase transitions. Note that the coexistence regions here are all small and have been simply presented as lines, similar to Refs. 6 and 7. In this paper we will explore the behaviour of point defects associated with the crystals that appear in this 2d phase diagram.

III. METHODS

A. General Simulation Details

We used Monte Carlo simulations in the NV T -ensemble with periodic boundary conditions [START_REF] Frenkel | Understanding molecular simulation: from algorithms to applications[END_REF] , where the particles interact via the point Yukawa potential (Eq. 1). The potential was truncated and shifted such that the shift was never more than 10 -5 k B T . The system size was chosen to always be large enough to accommodate this choice, such that the cutoff range is less than half the box length.

The system sizes were chosen depending on the phase in question. For studying defect concentrations we used systems containing between 250-1500 particles. For select points of the smallest systems we examined whether doubling the system size mattered, and in all cases it had no discernible effect on the defects concentrations. For studying the shape of the defects, we used system ranging from 1000-3500 particles and again ensured that the system size was not affecting the results.

B. Concentration of defects

To determine the vacancy concentration, we make the assumption that the defect concentration is sufficiently low that i) the defects are not interacting, ii) the equation of state of the crystal is unaffected by the presence of defects. In this case, the free energy of a system of N particles in a volume V at temperature T , can be written

β F vac (N,V, T ) = β M f df (N,V, T ) + β (M -N) f vac +N log N M + (M -N) log M -N M , (3)
where M > N is the number of lattice sites, f df is the free energy per particle of the defect-free crystal, and f vac is the free energy associated with creating a single vacancy at a specific lattice site.

Taking the Legendre transform to turn this Helmholtz free energy into a Gibbs free energy, and minimizing with respect to the number of lattice sites M, we find that the equilibrium concentration of vacancies is given by

n vac ≡ M -N N = exp[-β µ vac ], (4)
where µ vac is defined as f vac (ρ M , T ) + µ df (P, T ) with µ df (P, T ) the chemical potential of a defect-free crystal and ρ M = M/V the density of lattice sites. Note that P is the pressure.

For a crystal containing interstitials, where M < N, we use a similar approach yielding an equilibrium concentration of interstitials

n int ≡ N -M N = exp[-β µ int ], (5)
with

µ int = f int (ρ M , T ) -µ df (P, T ). Here f int (ρ M , T )
is the free energy associated with creating an interstitial at a specific lattice site. In order to obtain the concentration of point defects for various points along the phase boundary of the Yukawa crystal, we thus need to measure f vac (ρ M , T ), f int (ρ M , T ) and µ df (P, T ) in a Yukawa crystal. Because it is not possible to measure these free energies directly in a Monte Carlo simulation, we use thermodynamic integration as described below.

To obtain the chemical potential µ df of the defect-free crystals, we first use the Frenkel-Ladd method [START_REF] Frenkel | New Monte Carlo method to compute the free energy of arbitrary solids. Application to the fcc and hcp phases of hard spheres[END_REF] to obtain the Helmholtz free energy. We then combine this with the pressure, measured via the virial expression [START_REF] Frenkel | Understanding molecular simulation: from algorithms to applications[END_REF] , to determine the chemical potential.

We now turn our attention to the method for finding the free energies f int (ρ , T ) and f vac (ρ M , T ), associated with creating point defects. In the case of a vacancy, we break f vac up into two contributions: f vac = f shrink + f remove , where f shrink is associated with turning one of the particles of a defect-free crystal into a non-interacting particle, and f remove is associated with removing this non-interacting particle. Similarly, in the case of an interstitial we first compute the free energy, f add , associated with inserting a non-interacting particle and then calculate the free energy, f grow , associated with turning this non-interacting particle into a normal-interacting particle. Note that in all cases, the particle associated with a defect is confined to a single Wigner-Seitz cell.

We then calculate the total free energy for the interstitial using f int = f grow + f add . The free energies associated with f add and f remove are given by:

f add = -k B T ln V W S Λ 3 . (6)
and

f remove = k B T ln V W S Λ 3 , (7)
where V W S is the volume of the Wigner-Seitz cell and Λ is the thermal DeBroglie wavelength.

To calculate f shrink , we use thermodynamic integration with an auxiliary Hamiltonian

U λ = (1 -λ )U 0 + λU non-int , (8)
with U 0 the normal interaction potential of our system, and U non-int the potential energy of a system where one particle is non-interacting. Following standard thermodynamic integration, the free-energy difference between a crystal with one non-interacting particle and a defect-free crystal is then given by,

f shrink = F non-int -F df = 1 0 dλ U non-int -U 0 λ , (9)
with F non-int the Helmholtz free energy of a crystal containing one non-interacting particle. The ensemble average, ... λ is evaluated using the auxiliary potential given in Eq. ( 8). The free energy f grow is determined following the same method. Note that in both cases, we evaluate the integral numerically using 34 different values of λ . However, while in theory this method works fine, in practice the sampling can become very slow. When the system is at λ = 0 we are evaluating the energy difference in Eq. 9 using the potential U λ =0 = U non-int . This means that without any energy penalty, the non-interacting particle can come very close to other particles if those particles are near their Wigner-Seitz cell boundary. Because we compute the potential energy difference with the system where our particle does have interactions, the term U non-int -U 0 λ =0 can become very largein the interstitial case even infinitely large. Due to these large energy differences, the simulation needs a long time to get a reliable answer for the value U non-int -U 0 λ =0 .

To circumvent this problem we alter the potential. Instead of letting it diverge at r = 0 as it normally would do, we assume that the potential increases linearly below a certain defined value r alter . By doing so the potential has a finite value, U max , at r = 0. If we evaluate U non-int -U 0 λ =0 for higher λ 's this altered potential will not have any influence as the particles will never have a distance r with r < r alter due to the energy penalty. However for small λ , we avoid the large energies. Because in the end we integrate over the energy difference (see Equation ( 9)), this alteration to the potential has no influence on the final free energy. We find that the precise values for U max and r alter do not matter, as long as we make sure r alter is sufficiently small that for higher values of λ , r will almost never be smaller than r alter . We checked this by running the same simulation twice for different values of U max and r alter .

IV. RESULTS

We start our investigation by exploring the equilibrium concentration of vacancies and interstitials in both crystals. As a starting point we focus on state points in the vicinity of the fluid-crystal phase boundary -the region on the phase diagram that is expected to have the highest concentration of defects. To predict these concentrations we make the assumption that the defects do not interact, and that their effect on the pressure of the system is negligible. We can then use a combination of Monte Carlo simulations and thermodynamic integration to extract the defect concentrations. The result is shown in Fig. 1b 50 .

Clearly, along the fluid-crystal line, BCC appears to have more defects of both types than FCC. More specifically, for BCC both the vacancy and interstitial concentrations are on the order of 10 -4 , while for FCC the concentrations are closer to 10 -5 . For vacancies, these concentrations are similar to those found for hard spheres at coexistence (10 -4 ) [START_REF] Pronk | Point defects in hard-sphere crystals[END_REF] . However, the interstitial concentration in both cases is orders of magnitude higher than the 10 -8 predicted for interstitials in hard-sphere crystals at the fluid-crystal phase boundary [START_REF] Pronk | Point defects in hard-sphere crystals[END_REF] .

To make a more direct comparison of the behaviour of the two crystals, we then calculated the defect concentrations along the FCC-BCC phase line. From Fig. 1c we observe again that BCC generally has more defects than FCC; while the difference in the interstitial concentration is small, the difference in vacancy concentration varies from two to eight orders of magnitude. Clearly, BCC generally exhibits more equilibrium point defects than FCC.

To explore the large differences we observe between FCC and BCC, we now turn our attention to the structure of the point defects in these two crystals. To determine the structure, we performed NV T MC simulations with a single vacancy or interstitial present. To prevent the defect from hopping during our analysis, we confined all particles to their Wigner-Seitz cells [START_REF] Van Damme | Phase and vacancy behaviour of hard "slanted" cubes[END_REF][START_REF] Van Der Meer | High antisite defect concentrations in hard-sphere colloidal laves phases[END_REF] and then measured the average location of each particle during the simulation.

The results for a vacancy in FCC are shown in Fig. 2 ab, while e-f depict the average deformation associated with a vacancy in a BCC crystal. In both crystals, as expected, the largest deformation is associated with the first shell of neighbours -however, it is approximately twice as large in the case of the BCC crystal (note the different scaling on the color bars). This likely arises due to the lower number of particles in the first shell of BCC, 8 in comparison to the 12 in FCC. In the BCC crystal, the particles are less strongly caged by their neighbors, and hence more free to move into the space opened up by the vacancy. More interesting is the behaviour of the second shell of neighbours. While in FCC all particles again deviate in the direction of the vacancy, in BCC half the particles move towards the vacancy while the other half move away from the defect. The end result is a much larger change in energy of the crystal upon removing the particle: in the system shown in Fig. 2, the average difference is β ∆U FCC = -35.6 and β ∆U BCC = -42.9 in the FCC and BCC crystals, respectively. This difference contributes directly to the difference in free-energy cost for creating a vacancy in the two crystals. In short, the BCC structure is better able to take advantage of the vacancy to reduce its local potential energy, which helps to alleviate the cost of creating a vacancy and hence makes them more prevalent.

The interstitials turn out to be an even more interesting case. In Fig. 2 c-d, and g-h we plot the average displacement of particles from their lattice sites in FCC and BCC crystals, respectively. In comparison to the vacancy case, these defects appear at first rather similar: in both crystals the average deviations due to the interstitial are mainly along the lines pointing along the nearest-neighbour directions, and decay slowly through a number of neighbouring shells. In the FCC crystal this means that particles lying along the six 110 lines are displaced the most, similar to what was found in hard-sphere crystals [START_REF] Van Der Meer | Diffusion and interactions of point defects in hard-sphere crystals[END_REF] . In the BCC crystals it is the particles lying along the four 111 lines.

Interestingly, however, the averaged displacements do not tell the full story. If we examine the instantaneous realization of an interstitial, there is a spontaneous symmetry breaking in the displacement of neighboring particles along different directions, especially in the case of BCC. One clear way of demonstrating this is to quench a system containing an interstitial to high values of Γ, such that the system minimizes its potential energy. As shown in Fig. 4, this quench has a particularly remarkable effect on the BCC crystal. While in the case of FCC (Fig. 4a), the defect takes on a normal dumbbell structure with a 3d displacement field around it, in BCC the defect becomes one-dimensional (Fig. 4b): only particles along one of the four 111 lines are displaced significantly. With this knowledge in mind, we can recalculate the average displacement field for an interstitial by first rotating each configuration so that the defect is always oriented along the same axis. The result is shown in Fig. 4 for the quenched system and in Fig. 3 for the system at finite Γ.

This one-dimensional configuration strongly resembles a so-called crowdion: an exotic 1d defect proposed to exist in some metallic BCC crystals [START_REF] Paneth | The mechanism of self-diffusion in alkali metals[END_REF] . In order to characterize the structure of the defects, we measure the average particle displacements u n = x na 111 n near the interstitials along the defect direction, where x n is the position of particle n along the defect, and a 111 is the crystal lattice spacing along the 111 direction. We choose n = 0 to correspond to the particle just before the defect center and use "standard" boundary conditions: u n=-∞ = a 111 , u n=∞ = 0. We plot this displacement field for a BCC crystal with κ = 3.5 and Γ = 2400 in Fig. 3, along with the displacement field along the three other 111 lines. Clearly, particle positions along the defect direction are strongly affected by the presence of the interstitial, while along the other 111 directions they remain essentially unperturbed -indicating that the defect is one-dimensional.

A classic characteristic of a crowdion is that the defect shape can be well captured by the Frenkel-Kontorova model [START_REF] Kontorova | On the theory of plastic deformation and twinning[END_REF][START_REF] Dudarev | Coherent motion of interstitial defects in a crystalline material[END_REF] . This model described a one dimensional chain of particles that are connected to their neighbours via springs, and embedded in a periodic external potential. Defects are included as missing or extra particles with respect to the number of external periodic wells. In the continuum limit, the average particle positions near a defect follow the soliton so-

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▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ lution to the sine-Gordon equation, which has a single free parameter that captures the extent of the defect. Hence, to further confirm that the interstitials are realizations of crowdions we compare our results to the soliton solution of the sine-Gordon equation (black dashed line in Fig. 3b), using the extension of the defect as a fit parameter. We observe excellent agreement. Curious as to how the shape of the defect is dependent on where we are on the phase diagram, we performed the same analysis for a range of different state points with different values of κ and Γ. Remarkably, in the area of the phase diagram close to the melting line, we see very little effect of either parameter on the structure of the crowdion. In Fig. 3c, we show the displacement fields along the defect axis for 15 different state points, and obtain essentially the same curve every time. This indicates that in this regime, the extension of the defect is largely independent of both the interaction strength and screening parameter. We note, however, that the crowdions do slowly get longer as Γ is increased far beyond the melting point. This can be observed from the shape of the quenched defects, as shown in Fig. 4d.

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V. CONCLUSIONS

In summary, we have characterized the point defects that appear in crystals of charged colloids, an archetypical colloidal model system. Surprisingly, we found dramatic differ-ences between the two -fairly similar -crystal phases FCC and BCC. As a first observation, the BCC crystals contain dramatically more vacancies, as well as more interstitials, than their FCC counterparts. One logical explanation for this is the relatively small number of nearest-neighbors in the BCC crystals, which makes it easier for particles to partially emerge from their cages and make use of the extra room opened up by a vacancy -or adapt to the encroachment of a nearby interstitial.

BCC not only exhibits significantly more defects, but its interstitials manifest as exotic one-dimensional defects called crowdions [START_REF] Paneth | The mechanism of self-diffusion in alkali metals[END_REF] . The delocalized nature of such defects would be expected to promote fast and strongly anisotropic diffusion of the defects through the crystal [START_REF] Derlet | Multiscale modeling of crowdion and vacancy defects in body-centered-cubic transition metals[END_REF][START_REF] Han | Self-interstitials in V and Mo[END_REF][START_REF] Zepeda-Ruiz | Strongly non-arrhenius self-interstitial diffusion in vanadium[END_REF][START_REF] Dudarev | Coherent motion of interstitial defects in a crystalline material[END_REF] . In combination with the relatively large concentration of interstitials in BCC near melting, these defects are expected to strongly impact the transport properties of the crystal, including self-diffusion and the diffusion of dopants [START_REF] Tauber | Anomalous dynamics of interstitial dopants in soft crystals[END_REF] .

The observation of a crowdion defect in an easy-to-realize and highly tunable colloidal system is not only important for our understanding of this system itself, but also for understanding the nature of crowdions. In atomic systems, crowdions are both rare and hard to observe directly (almost all studies are based on simulations and theory [START_REF] Derlet | Multiscale modeling of crowdion and vacancy defects in body-centered-cubic transition metals[END_REF][START_REF] Nguyen-Manh | Self-interstitial atom defects in bcc transition metals: Group-specific trends[END_REF][START_REF] Osetsky | Onedimensional atomic transport by clusters of self-interstitial atoms in iron and copper[END_REF][START_REF] Han | Self-interstitials in V and Mo[END_REF][START_REF] Zepeda-Ruiz | Strongly non-arrhenius self-interstitial diffusion in vanadium[END_REF][START_REF] Dudarev | Coherent motion of interstitial defects in a crystalline material[END_REF] ). In contrast, charged colloids can be studied in real space and real time using e.g. confocal microscopy, and hence are an ideal experimental playground for studying these defects. Surprisingly, despite decades of intense study, it appears that this fundamental colloidal model system has not yet given up all of its secrets.

FIG. 1

 1 FIG. 1. a) Phase diagram of Yukawa systems in the (κ, Γ) plane, with the phase boundaries from Refs. 6 and 7. Vacancy and interstitial concentrations plotted along b) fluid-crystal phase boundaryand c) BCC-FCC phase boundary. We estimate an error of up to 0.2k B T in our calculation of the µ vac(int) , leading to an approximate error of a factor of 1.2 in the concentrations.

FIG. 2 .

 2 FIG. 2. Average lattice deformation due to a point defect in a-d) the FCC crystal at κ = 3.5 and Γ = 2565 and e-h) the BCC crystal at κ = 3.5 and Γ = 2400. a-b,e-f) Deformation due to a vacancy (indicated by red or a dotted sphere) and c-d,g-h) deformation due to an interstitial (indicated by a black sphere). Left: 3d representation of part of the simulation box. The gray points represent the lattice sites. Right: projection of the displacement vectors on two (100) planes on top of each other. The black points represent the lattice sites that lie in the plane of the defect and the gray points the lattice sites in the neighboring plane. In all figures the size of the arrows is exaggerated, but the color of the arrows indicates the deformation in terms of the Wigner Seitz radius a.

5

 5

FIG. 3

 3 FIG. 3. a) Lattice deformation due to an interstitial in the BCC crystal at κ = 3.5 and Γ = 2400. The gray points represent the lattice sites of part of the simulation box and the black spheres represent the positions of the interstitial and its companion. The size of the arrows is exaggerated, but the color of the arrows indicates the deformation in terms of the Wigner Seitz radius a. Note that these deformations are averaged over multiple configurations which have been rotated so that the defect always points along the same direction. b) Displacement u n along the four 111 directions for the same system as a). The blue dots indicate u n along the direction of the crowdion and the dashed line represents the corresponding fitted soliton solution. c) u n along the direction of the crowdion for κ = 2.0 and Γ = 1000, 1100, 1205 (blue dots), κ = 2.5 and Γ = 1000, 1100, 1205 (yellow squares), κ = 3.0 and Γ = 1205, 1400, 1600 (green diamonds), κ = 3.5 and Γ = 2200, 2300, 2400 (orange triangles), κ = 4.0 and Γ = 4000, 4100, 4200 (purple triangles). The lines represent the corresponding fitted soliton solutions.

FIG. 4 .

 4 FIG. 4. Quenched lattice deformation due to an interstitial in a) the FCC crystal and b) the BCC crystal, both at κ = 3.5. The gray points represent the lattice sites of part of the simulation box and the black spheres represent the actual positions of the interstitial and its companion. The size of the arrows is exaggerated, but the color of the arrows indicates the deformation in terms of the Wigner-Seitz radius a. c) Displacement u n along the four 111 directions for the same system as b). The blue dots indicate u n along the direction of the crowdion and the dashed line represents the corresponding fitted soliton solution. d) u n along the direction of the crowdion for five different κ. The lines represent the corresponding fitted soliton solutions.

VI. ACKNOWLEDGEMENTS

The authors thank Alfons van Blaaderen, Anna Nikolaenkova, and Emanuele Boattini for many useful discussions. L.F. acknowledges funding from the Dutch Research Council (NWO) for a Vidi grant (Grant No. VI.VIDI.192.102). B.v.d.M. acknowledges funding from the Rubicon research program with project number 019.191EN.011, which is financed by NWO.

VII. DATA AVAILABILITY STATEMENT

The data that support the findings of this study are available from the corresponding author upon reasonable request.