Role of spherical particles on magnetic field recording in sediments: Experimental and numerical results

Physics of the Earth and Planetary Interiors 214 (2013) 1–13

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Role of spherical particles on magnetic field recording in sediments: Experimental and numerical results Dario Bilardello a,⇑, Josef Jezek b, Stuart A. Gilder a a b

Department of Earth and Environmental Sciences, LMU, Theresienstrasse 41, 80333 Munich, Germany Institute of Applied Mathematics and Information Technologies, Faculty of Science, Charles University in Prague, Albertov 6, 128 43 Praha 2, Czech Republic

a r t i c l e

i n f o

Article history: Received 15 May 2012 Received in revised form 5 September 2012 Accepted 31 October 2012 Available online 9 November 2012 Edited by Chris Jones Keywords: Paleomagnetism Inclination shallowing Deposition experiments Numerical models

a b s t r a c t We report deposition experiments using spherical glass beads that possess remanent magnetizations stemming from iron impurities. 15 g of glass beads with a well-characterized size distribution were loaded in two different sets of tubes with diameters of 2.0 and 3.6 cm. Each tube contains identical column heights of de-ionized water, thereby allowing us to assess the effect of sediment concentration on the results (352 versus 90 kg/m3 [g/l], respectively). The tubes were placed in magnetic fields of variable inclination and intensity in a temperature-controlled environment. The full vector magnetization and sediment accumulation rates were measured upon deposition times ranging from 10 min to 10 days. Experiments were run in triplicate to evaluate data reproducibility. Together with the lack of magnetic interaction and the absence of clumping, the experiments elucidate an end-member scenario of how sediments acquire remanent magnetizations in the absence of flocculation. Our results show that inclination shallowing, in the range of 7–20° for field inclinations of 30° and 60°, is indeed possible with solely spherical particles. More importantly, we observe a field dependence on the inclination error. Field dependence on the moment acquisition and inclination error both exhibit non-linearity, which may complicate interpretations of relative paleointensity data in paleomagnetic records. A newly developed numerical model, whereby particle collision during settling combined with both rolling and slipping (translation) on the substrate, is consistent with the experimental results. Ó 2012 Elsevier B.V. All rights reserved.

1. Introduction Acquisition of remanent magnetization in sediments, called a depositional remanent magnetization (DRM), is typically described by spherical magnetic particles falling through a stagnant water column (Rees, 1961; Collinson, 1965; King and Rees, 1966; Stacey, 1972; Tauxe and Kent, 1984; Shive, 1985; von Dobeneck, 1996; Katari et al., 2000). Viewed in this way, the particle is subject to balanced inertial, viscous and magnetic torques, and spherical magnetic particles attain perfect alignment with the ambient field within seconds (Nagata, 1961; Collinson, 1965). The situation is much more complicated in nature, where the sedimentation process spans a vast parameter space regarding particle size and shape distributions, viscosity, pH and Eh of the fluid, etc. (Verosub, 1977). Contact forces between particles and Brownian motion also play a role. Eventually the particles encounter the substrate, leading to mechanical interaction and possibly experiencing shear from bottom currents. Within the sediment column, bioturbation, dewatering, diagenesis and compaction can modify the magnetization, ⇑ Corresponding author. Current address: Instituto de Geociências, University of São Paulo, Rua do Lago 562, 05508-080 São Paulo, Brazil. E-mail address: [email protected] (D. Bilardello). 0031-9201/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.pepi.2012.10.014

which is known as a post-depositional remanent magnetization (pDRM) (see Tarling and Turner, 1999 and references therein). Laboratory redeposition experiments reveal that the magnetization intensity of sediments grows in proportion to the strength of the applied field and that the net magnetization is orders of magnitude lower than the saturation remanence (i.e., if all the particle moments were parallel) (e.g., Barton et al., 1980; Tauxe and Kent, 1984). Several experiments demonstrate that the net effect of a depositional remanent magnetization is to shallow the remanent inclination in the rock (IR) with respect to the applied field inclination (IB) such that tan(IR) = f  tan(IB), where f is the flattening factor (King, 1955; Løvlie and Torsvik, 1984; Tauxe and Kent, 1984). Misalignment of declination is negligible. Two basic models are used to explain inclination shallowing. In that of King (1955), sediments are composed of spherical and platy particles: shallowing depends on the relative contribution of the latter. Griffiths et al. (1960) explained inclination shallowing by equal-sized spherical particles rolling into depressions between grains lying on the sedimentation plane. On the other hand, however, instances of natural sediments yielding inclinations compatible with those predicted from apparent polar wander paths and possessing the same inclinations as lava flows, which are mostly immune to inclination shallowing have also been reported (e.g., Opdyke, 1961).

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Several experimental and theoretical studies on DRM acquisition have focused on particle aggregation (flocculation) during settling, which affects the magnetic intensity and inclination recorded by sediments based on the clay content, clay mineralogy and fluid conductivity (Ellwood, 1979; Shcherbakov and Shcherbakova, 1983; Lu et al., 1990; Deamer and Kodama, 1990; Sun and Kodama, 1992). For example, Sun and Kodama (1992) found that magnetic grains attach to clay minerals by either electrostatic or van der Waals forces. The magnetic grains become incorporated into the clay fabric of the sedimentary rock and rotate with the clay particles during post-depositional compaction (Arason and Levi, 1990; Katari and Bloxhamm, 2001). Van Vreumingen, 1993a,b) showed that flocculation varies as a function of salinity of the sediment suspension, and Tauxe et al. (2006) found a non-linear field dependence on remanence for certain floc sizes. Mitra and Tauxe (2009) explored remanence acquisition as a function of applied field and floc size distributions. Their work helped explain discrepancies in relative paleointensity and inclination data, highlighting the complex nature of DRM acquisition with respect to different sedimentary environments (variable salinity, mineralogy, organic matter content, etc. (see also Katari and Tauxe, 2000). Shcherbakov and Sycheva (2008, 2010) recognized that more than seven parameters are needed to describe the magnetization acquisition of sediments. This multi-parametric control on DRM hinders relative paleofield intensity estimates by redeposition methods since laboratory conditions do not reproduce the natural environment. Other workers have addressed the question of lock-in depth of the magnetic signal in sediments (Kent, 1973; Tucker, 1980; Bleil and von Dobeneck, 1999; Roberts and Winklhofer, 2004). For example, Løvlie (1974, 1976) attributed the lag between changes in ambient field and lock-in of the magnetization to post-depositional alignment of the magnetic grains, with consolidation-rate significantly influencing a sediment’s magnetic intensity. Irving and Major (1964) showed that sediments first deposited in a null field, and then subjected to an applied field, accurately recorded the field direction. While post-depositional effects seem to play a role in influencing the final orientation of the magnetic vector, other laboratory experiments and numerical models find a limited influence (Verosub et al., 1979; Shcherbakov and Shcherbakova, 1987; Katari et al., 2000). Despite significant efforts to understand the genesis of a detrital remanent magnetization, complete knowledge of the underlying principles are still lacking. We thus initiated a series of experiments to focus on a particular aspect of the problem—namely the contribution of solely spherical grains in the absence of flocculation. Our experiments do not intend to simulate a natural detrital remanent magnetization acquisition, rather to unravel one specific factor that contributes to it. Hence, we carried out deposition experiments with synthetic spherical magnetic particles whose

size distribution is well known. Numerical simulations are developed to explain the results. We are particularly interested whether spherical particles can produce inclination shallowing by rolling on the substrate when they settle. This necessitates a re-evaluation of the classical rolling spheres model of Griffiths et al. (1960). Our experimental results and numerical models contradict the idea of King (1955) that spherical particles accurately record the ambient field direction,

2. Materials and methods 2.1. Glass beads Fig. 1a shows a scanning electron microscope image of the solid glass spheres (Potters Europe, spheriglass 5000) used in this study. The image attests that almost all the particles are spherical in shape. Laser particle analysis (Coulter) was used to measure the grain-size spectra of the beads (Fig. 1b). Five independent runs, made without using dispersing agents or ultrasound, are highly reproducible and reveal no evidence for clumping or clustering of the particles. Particle radii range between a fraction of a micrometer to 11.4 lm, showing a sharp peak for the smallest grain sizes, then decreases almost exponentially with increasing radii. Overall, 10% of the particles have radii <0.48 lm, 25% are <1 lm, 50% are <3 lm, 75% are <6.8 lm and 90% are <7.7 lm. Company specifications list a median radius of 1.7–3.5 lm, with 90% of the spheres having radii between 0.3 and 9.7 lm. We accepted the company’s reported density of 2.5 g/cm3 without independent verification. Experiments on the beads indicate they contain impurities that carry a magnetic remanence. Hysteresis loops, backfield curves and magnetization versus temperature curves were measured with a Petersen Instruments, variable field translation balance (dwell field 30 mT, dwell time 1 s, ramp slope [heating and cooling] 40 °C/min). The magnetic moment upon heating undergoes a change in slope at 580 °C followed by a major drop around 770 °C, suggesting the presence of both magnetite and iron, respectively (Fig. 2a). Magnetic intensities during cooling lie systematically below those of heating, signifying a net loss of magnetic moment during heating. Repeat heating–cooling cycles in 100 °C intervals indicate that already below 400 °C the cooling curve is lower than the heating curve. Our interpretation is that the glass contains only pure Fe as a remanence carrier, but that Fe partly oxidizes into magnetite during heating. Acid rinsed, non-annealed glass beads were used in the experiments. Fig. 2b shows a typical hysteresis loop of the material measured at room temperature. Hysteresis loops determined on four independent samples exhibit a high degree of reproducibility, with hysteresis parameters lying within the pseudo-single domain field on

Fig. 1. (A) SEM image of the glass beads used in the deposition experiments highlighting the size distribution and the predominantly spherical shape of the particles. (B) Grain-size distribution of the particles measured with a laser counter (no de-flocculants or ultrasound used). The grain-size bins sum to 100%.

D. Bilardello et al. / Physics of the Earth and Planetary Interiors 214 (2013) 1–13

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Fig. 2. Summary of magnetic measurements conducted on the glass beads. (A) Thermo-remanent curves for the heating (red) and cooling (blue) cycle between room temperature and 800 °C measured with a Petersen Instruments, variable field translation balance. 580 and 770 °C correspond to Curie temperatures of magnetite and pure iron, respectively. (B) A magnetic hysteresis loop measured with a Petersen Instruments, variable field translation balance. (C) Hysteresis parameters from four independent measurements lie within the pseudo-single domain field on a Day et al. (1977) plot, although this is not strictly valid for iron metal. (D) A first order reversal curve diagram measured with a Princeton Instruments, vibrating sample magnetometer. The diagram is representative of interacting single domain particles (Roberts et al., 2000). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

a Day et al. (1977) plot (Fig. 2c). Such plots were specifically developed for titanomagnetite and might not be applicable for iron. The average saturation magnetization (Ms) and remanent saturation magnetization (Mrs) are 3.46  103 A m2/kg (8.64 A/m) and 5.05  104 A m2/kg (1.26 A/m), respectively, while the bulk coercive force (Bc) and coercivity of remanence (Bcr) are 12 and 30 mT, respectively. First order reversal curves, measured with a Princeton Instruments vibrating sample magnetometer at the Institute for Rock Magnetism (Minneapolis) (settings: field range 300 mT, moment range 2  106 A m2, averaging time 800  103 s, analyzed using the code of Michael Winklhofer [version July, 2010]), indicate that the beads contain interacting single domain particles (Fig. 2d). In assigning a magnetic moment to the beads, we recall that the remanence ratio (Mrs/Ms) of a random assemblage of non-interacting single domain particles with uniaxial anisotropy is 0.5 (Dunlop and Özdemir, 1997). Mrs/Ms. of the glass beads is 0.146, or 3.4 times less than the expected value, which we attribute to magnetic interaction. Assuming that each bead contains a proportional amount of magnetic material by volume, and that the magnetic remanence of each bead corresponds to the saturation value, a magnetization of 8.64/3.4 = 2.5 A/m can be assigned to each individual bead. Moreover, after immersing the grains in water and observing them under a microscope, no clumping or aligning of the grains was detected. Thus, magnetic interactions revealed in Fig. 2d likely occurs within each grain (intra-grain) and not between grains (inter-grain), further supporting that 2.5 A/m is the most reasonable value for the magnetization value of the beads. We tested for magnetic viscosity by imposing a 600 mT field on the beads to saturate their magnetizations and then measured the

remanence of a sample over several hours after exposure. Remanence decreased by 2% in the first 19 h, after which it remained stable. The sample was then placed in a 50 lT field, orthogonal to the magnetization direction. No change in moment or direction was detected 17 h later, thus lending confidence that the material is suitable for the deposition experiments. 2.2. Experimental set-up and procedure Helmholtz coils, 1 m on each side, were constructed to generate a stable magnetic environment up to 100 lT in any direction over the volume in which the experiments were performed. A fluxgate magnetometer probe fixed within the coils constantly monitored the field. Non-magnetic, cylindrical, flat-bottomed borosilicate tubes with diameters of 2.0 and 3.6 cm (height = 30 cm) were made in triplicate for each tube size. Tubes of both diameters were prepared with 15.000 ± 0.001 g of dry glass beads; distilled water was added such that the height of the water column was 20 cm. The sediment mass is thus identical for both tube-sizes, yet the 2 cm diameter tubes have 3.9 times higher sediment concentrations than the 3.6 cm diameter tubes. In this way, the parallel experiments can help identify potential wall and concentration effects (i.e., hitting of the particles against the tube walls or mutual hitting of the particles as they settle). The experiments were carried out by placing the tubes in an ultrasound bath while shaking vigorously to achieve full separation and suspension of the sediment. The tubes were placed in the controlled magnetic environment within the Helmholtz coils at a specified field intensity and inclination. Temperature fluctuations that could potentially change the moment or water viscosity were

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minimized by immersing the tubes in a temperature-controlled water tank (32 °C) within the Helmholtz coils. For example, the experiments reported here were carried out for over a year’s time span where the temperature in the room fluctuated at least 10 °C. Water viscosity decreases 20% from 18 to 28 °C (Kestin et al., 1978), which would alter the falling time of the spheres in the tubes depending on when the experiments were done. After a given settling time, the tubes were removed from the Helmholtz coils and placed into a vertical, 3-axis, 2G Enterprises, 755 cryogenic magnetometer to measure the magnetic vector. Typical measured magnetic moments range from 7  107 to 4  106 A m2 (Table 1), which are well above the noise of the magnetometer over the measurement time (1–3  1011 A m2). The horizontal plane of the tubes was positioned in the magnetometer such that the middle of the fully deposited sediment was at the point of maximum signal strength on the magnetometer sensors. Settling intervals under a given field inclination and intensity were normally 10, 20, 30, 40, 50 and 60 min, then 2, 3, 4, 5, 18, 68 and 240 h. After each measurement step, the tubes were vigorously shaken, and then placed back into the Helmholtz coils for a different settling time. In this manner, each time interval can be considered as an independent experiment, also meaning that settling steps may be performed in any order and repeated if necessary. The fact that measurements for different time intervals yield rather smooth curves attests to the uniformity of the underlying processes and increases the reliability of the results. Three tubes of each diameter were run contemporaneously to examine data reproducibility. The thickness of the sedimentary column was measured after each time step to determine the volume of deposited sediment, which could be compared to the accumulated moment. This measurement also quantifies sediment packing and monitors the consistency of the experiments.

3. Results from the deposition experiments Figs. 3 and 4 show the results of all experiments conducted with the 2.0 and 3.6 cm diameter tubes, respectively (data in Table 1— figures and subsequent analyses exclude values in gray). Table 2 summarizes the results for each experiment by calculating the mean of the last three experimental steps for each of the three tubes, and then taking the average and single standard deviation of those three mean values. In some cases the moments or inclinations vary slightly in time until the last deposition step (Figs. 3 and 4, Table 1); however, taking only the last step instead of the average of the last three steps yields virtually identical results. In general, the data show a relatively good degree of reproducibility for all experiments carried out at a given applied field strength (B) and inclination (IB). That the curves of magnetization acquisition over time match the sedimentation rates lends credence to the results (Figs. 3 and 4); this will be explored in further detail in the numerical modeling section below. We calculated the slope of the average moments from the last three time steps reported in Table 2 (not shown), and found it to be is positive in 10 of 11 cases when considering data where the coefficient of determination (R2) from the least squares linear fit exceeds 0.85. This could suggest that the DRM acquisition process is not completed after 10 days of deposition. Alternatively, the change in slope between short (ca. <300 min) and long (ca. >300 min) time scales could reflect a different acquisition process with a DRM at short time scales and a pDRM over long time scales. A similar analysis for the height data leads to a similar conclusion as the moment data: the slope of the average heights from the last three time steps (data from Table 2) is positive in 7 of 8 cases where (R2) exceeds 0.85. Hence separating the data into two separate regimes over different time scales seems unlikely, e.g., DRM

continues, albeit much slower, even after 10 days of settling time. No systematic correlation is found between the slope of the moment acquisition and B. Initial loading of the tubes was in July 2010. The last experiments performed were at IB = 0° under B of 50 (March, 2011), 25 (April, 2011) and then 100 lT (May–June, 2011). During the experiments at 25 lT, 2.0 cm tube #3 displayed much higher sediment thicknesses and lower moments than the other two tubes (Table 1, data in gray). In the subsequent experiments conducted at IB = 0°, B = 100 lT, all 2.0 cm tubes began to mimic this behavior, so the experiments at 100 lT were discontinued. We think the glass beads started welding together, which would lower the magnetization and increase the pore space of the deposited material, thereby increasing the thickness of deposited sediment. This highlights the usefulness of measuring the sediment accumulation rate as an independent control. Seen at a given field intensity, the magnetization acquisition curves are compatible, independent of field inclination, and the moments are systematically higher with greater applied fields at a given time step. All curves show an initial steep rise in moment before flattening out after 100 min (2.0 cm tubes) and 300 min (3.6 cm tubes). Initial moment accumulation is thus faster in the 2.0 cm tubes, which makes sense, as the concentration is higher. Remanent inclination (IR) is fairly constant during the entire progression of the experiments, with only one exception at IB = 60°, B = 100 lT in the 3.6 cm tubes. This is a first-order observation since the process leading to inclination shallowing occurs immediately in the deposition process. It is readily apparent that for IB of 30° and 60°, IR varies as a function of B. One exception is for IB of 30° in the 2.0 cm tubes where IR is 23° for B of both 50 and 100 lT; however, IR is 15° when B = 25 lT. Extrapolating more specific trends out of the data becomes speculative as certain features are not coherent at given IB and B among both tube diameters; e.g., the greatest variability in IR (and sediment height) occurs when IR is 0° in the small tubes yet higher scatter is not observed in the large tubes.

4. Discussion 4.1. Magnetic moment Fig. 5a plots the mean magnetic moment of the last three measurement steps for each tube, averaged over the three settling tubes, versus applied field intensity (B). To a first-order, the fielddependence on the acquired magnetization is evident in all experiments, in conformity with virtually all experiments performed to date (e.g., King, 1955; Tauxe and Kent, 1984; Quidelleur et al., 1995). Tauxe and Kent (1984) performed deposition experiments from disaggregated red beds, which contained both magnetite and hematite. They placed approximately 10 gm of dry sediment in a 3.5 cm-diameter plastic tube and added water to make a 15 cm high slurry. This sediment to water concentration is comparable to our 3.6 cm diameter tubes. The tube was sealed, agitated and then placed in a controlled field for about 5 h. Both our results and those of Tauxe and Kent (1984) show more variability at the highest applied field (100 lT) over the range of inclinations (Fig. 5a), yet we find the two data sets are relatively compatible. Tauxe and Kent (1984) inferred a linear relationship between applied field and magnetization. Fig. 5a offers an alternative interpretation that the increase in moment is steeper from 25 to 50 lT than from 50 to 100 lT, which suggests the evolution can be better described by a non-linear (logarithmic) function. On the other hand, the small diameter tubes with higher sediment concentrations give more linear trends than the large diameter tubes. This could sug-

Table 1 Moment, inclination and thickness data measured in this study. Data are subdivided into three measurement tubes of 2.0 and 3.6 cm diameter with field inclinations of 60°, 30° and 0° subdivided into applied field intensities of 25, 50 and 100 lT. Data in gray were not considered for analysis (see text for details); experiments in horizontal fields of 100 lT were terminated after two time steps. Time (min)

2.0 cm diameter tubes

3.6 cm diameter tubes

Tube 1

Tube 2

Tube 3

Tube 1

Moment Inclination Height (A m2  10-6) (°) (mm)

Moment Inclination Height Moment Inclination Height (A m2  10-6) (°) (mm) (A m2  10-6) (°) (mm)

Tube 2

Tube 3

Moment Inclination Height Moment Inclination Height (A m2  10-6) (°) (mm) (A m2  10-6) (°) (mm)

Moment Inclination Height (A m2  10-6) (°) (mm)

8.5 11.3 16.5 25.0 31.5 33.2 34.9 36.1 36.6 36.3

1.18 1.38 1.64 1.84 2.02 1.99 2.15 2.15 2.13 2.24

43.2 45.7 41.7 42.6 37.7 40.9 41.5 38.6 36.7 37.3

10.3 13.6 18.7 26.7 32.6 34.2 35.9 37.3 38.3 37.6

0.67 1.16 1.48 1.90 2.11 2.08 2.19 2.29 2.26 2.39

37.8 38.9 37.9 38.3 39.6 40.2 38.2 43.0 45.3 40.0

3.1 4.6 5.3 7.5 8.8 9.2 9.9 9.1 9.2 10.2

0.76 1.23 1.37 1.80 1.92 2.04 1.99 2.13 2.14 2.32

38.5 38.4 41.6 38.5 39.3 38.9 39.1 41.8 42.6 41.7

3.3 4.8 5.9 7.6 8.6 9.3 10.1 9.3 9.2 10.5

0.92 1.28 1.51 1.85 2.01 2.14 2.20 2.27 2.26 2.49

35.3 36.1 38.0 37.3 38.4 37.6 38.3 41.0 42.0 40.6

4.0 5.2 6.4 7.8 9.2 9.6 10.0 10.2 10.2 10.2

Laboratory applied field and inclination values: 50 lT, 60° 10 1.32 50.4 7.5 1.10 49.1 20 2.20 48.4 13.0 2.09 49.9 30 2.59 50.3 17.7 2.29 48.9 60 3.00 47.5 25.6 2.95 48.3 120 3.13 48.0 30.9 2.95 48.0 180 3.20 48.2 33.1 2.98 48.3 240 3.22 49.4 34.2 3.00 47.5 1020 3.26 49.5 36.7 3.01 48.1 4080 3.30 49.9 36.8 3.08 48.0 14400 3.35 47.8 36.9 3.13 47.9

8.2 13.3 18.0 25.4 31.2 33.4 34.5 37.3 37.4 37.2

1.10 2.07 2.55 2.81 2.96 3.03 3.05 3.09 3.15 3.18

50.6 50.5 51.0 49.2 48.4 49.3 49.8 48.9 48.8 49.1

8.4 14.3 18.6 26.3 31.9 34.1 35.1 37.5 37.4 37.5

1.29 2.18 2.50 3.00 3.20 3.26 3.29 3.35 3.40 3.47

44.9 45.5 45.9 44.2 45.5 46.5 46.9 47.7 46.6 47.9

3.4 5.4 6.5 7.7 8.7 9.0 8.8 8.7 9.4 9.9

1.18 1.92 2.34 2.71 2.90 3.01 3.06 3.17 3.17 3.37

43.7 45.9 44.4 44.1 45.1 45.0 46.5 45.9 45.9 45.7

3.2 5.7 6.4 8.5 9.0 9.4 9.4 9.8 10.3 10.4

1.24 2.13 2.50 2.98 3.22 3.27 3.34 3.45 3.47 3.57

43.2 43.8 44.1 43.3 44.9 44.9 46.0 46.2 45.9 47.1

3.2 5.3 6.2 8.3 9.0 9.2 9.5 9.9 10.3 10.4

Laboratory applied field and inclination values: 100 lT, 60° 10 1.38 50.1 6.8 1.06 51.1 20 2.63 52.4 12.3 2.19 53.9 30 3.20 53.5 16.9 2.92 53.2 60 3.86 52.5 25.2 3.68 52.2 120 4.24 52.4 30.8 3.90 51.7 180 4.23 52.6 32.8 3.90 51.9 300 4.29 52.2 33.7 4.03 51.1 1020 4.58 53.0 36.1 4.08 52.4 4080 4.46 52.9 36.2 4.04 51.3 14400 4.70 49.8 36.6 4.27 51.1

7.0 12.6 16.6 25.3 31.0 32.9 34.5 36.3 36.8 36.9

1.30 2.56 3.04 3.78 4.04 4.05 4.19 4.17 4.27 4.37

53.1 55.7 54.9 52.1 51.5 51.7 51.5 53.5 51.5 52.5

7.2 13.7 18.2 26.3 31.8 33.3 35.1 36.5 36.9 37.1

1.33 1.90 2.78 3.41 3.74 3.85 4.20 4.45 4.46 4.69

43.6 42.1 46.3 49.6 50.6 50.8 50.5 51.1 51.8 51.2

3.5 5.1 6.5 7.5 9.0 9.6 9.9 10.1 10.0 10.6

0.98 1.85 2.25 3.25 3.54 3.66 3.86 3.96 4.02 4.10

43.1 45.6 44.0 48.4 50.2 50.3 50.7 50.5 51.3 50.4

3.1 5.1 5.8 8.0 9.3 9.7 9.8 10.1 10.3 10.4

1.15 1.92 2.88 3.40 3.74 3.96 4.31 4.47 4.45 4.64

40.0 42.9 47.2 48.5 49.6 49.9 50.1 50.1 50.5 49.7

3.2 5.3 6.4 7.9 9.4 10.1 10.2 10.5 10.7 10.9

Laboratory applied field and inclination values: 25 lT, 30° 10 0.69 15.9 4.6 0.83 11.6 20 1.76 14.3 15.6 1.76 13.8 30 2.03 14.8 18.6 1.91 12.7 60 2.30 15.1 25.7 2.13 13.1 120 2.46 15.3 31.3 2.30 13.3 180 2.51 15.6 33.5 2.34 15.6 300 2.58 17.5 35.6 2.40 14.0 1020 2.60 15.9 36.3 2.43 13.7 4080 2.59 16.3 37.3 2.41 14.0 14400 2.61 21.0 36.5 2.46 17.9

7.0 16.5 19.8 26.5 31.5 33.5 35.2 36.8 37.6 36.9

1.15 1.82 1.93 2.19 2.24 2.39 2.39 2.47 2.58 2.67

14.8 15.1 16.8 14.7 13.6 13.2 14.8 13.6 12.8 17.6

9.9 20.1 24.0 29.9 34.5 35.7 37.5 38.9 39.3 37.4

0.69 1.47 1.74 2.10 2.36 2.42 2.51 2.43 2.16 2.44

17.2 18.1 18.4 19.6 18.5 18.8 18.3 19.1 17.3 17.2

2.9 4.9 5.5 7.6 7.8 8.1 9.5 10.6 10.6 10.5

0.82 1.44 1.70 1.96 2.16 2.31 2.36 2.45 2.45 2.34

17.4 17.8 17.4 18.9 19.0 17.8 18.3 18.1 17.3 16.5

3.2 5.0 5.9 7.5 8.5 8.5 9.1 9.9 9.9 10.6

1.02 1.59 1.78 2.12 2.38 2.46 2.53 2.56 2.56 2.62

16.5 16.4 16.8 17.1 17.5 17.7 17.6 17.8 17.7 17.7

3.6 5.5 6.5 7.6 8.9 9.2 9.9 10.7 10.5 10.9

Laboratory applied field and inclination values: 50 lT, 30° 10 1.12 21.3 5.9 1.34 22.2 20 2.35 22.4 11.8 2.25 22.2 30 3.03 24.0 18.6 2.84 22.8

7.5 13.2 21.3

1.78 2.60 3.09

22.2 23.1 22.0

9.0 15.6 22.8

1.51 2.17 2.34

18.9 19.5 19.9

3.7 5.6 6.3

1.06 1.79 2.25

17.9 18.9 18.5

3.3 5.1 6.7

0.86 1.71 2.40

17.1 19.1 19.0

2.6 4.9 6.5

D. Bilardello et al. / Physics of the Earth and Planetary Interiors 214 (2013) 1–13

Laboratory applied field and inclination values: 25 lT, 60° 10 0.90 42.8 7.1 0.97 38.5 20 1.22 40.5 9.8 1.19 38.9 30 1.60 42.9 15.5 1.52 38.4 60 1.92 41.6 24.8 1.77 38.7 120 2.10 40.7 31.4 1.91 37.6 180 2.22 45.1 32.8 2.08 42.1 300 2.14 41.2 34.1 2.00 39.2 1020 2.22 42.6 35.5 2.06 39.4 4080 2.23 39.3 36.5 2.05 37.7 14400 2.26 43.0 36.3 2.11 42.8

(continued on next page) 5

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Table 1 (continued) Time (min)

60 120 180 240 300 1080 4080 14400

2.0 cm diameter tubes

3.6 cm diameter tubes

Tube 1

Tube 2

Moment Inclination Height (A m2  10-6) (°) (mm)

Moment Inclination Height Moment Inclination Height (A m2  10-6) (°) (mm) (A m2  10-6) (°) (mm)

Moment Inclination Height Moment Inclination Height (A m2  10-6) (°) (mm) (A m2  10-6) (°) (mm)

Moment Inclination Height (A m2  10-6) (°) (mm)

3.19 3.46 3.51 3.51 3.54 3.57 3.66 3.72

2.87 3.14 3.20 3.20 3.25 3.26 3.34 3.47

22.6 24.7 23.9 23.1 24.2 23.6 24.2 22.3

23.0 30.2 32.5 33.9 34.7 35.5 36.2 36.5

Tube 3

Tube 1

Tube 2

Tube 3

23.6 30.7 32.7 33.9 34.1 35.6 36.4 36.4

3.15 3.39 3.42 3.46 3.45 3.49 3.61 3.79

21.6 21.3 20.9 21.6 22.2 21.9 23.8 21.9

25.3 31.4 33.4 34.0 35.0 36.2 36.8 36.8

2.81 3.06 3.21 3.05 3.11 3.32 3.49 3.52

19.5 19.8 20.2 19.6 19.0 21.2 20.6 21.2

7.7 9.0 9.0 9.7 9.9 9.6 9.7 9.6

2.59 2.78 2.84 2.86 2.90 3.04 3.23 3.30

18.5 19.7 19.8 19.0 18.4 19.6 20.3 19.9

7.9 9.2 9.5 9.9 10.3 9.9 9.8 9.8

2.81 3.03 3.12 3.19 3.11 3.30 3.57 3.59

19.1 19.7 19.6 19.4 18.6 20.7 21.0 21.4

8.4 9.1 9.4 10.1 10.4 10.3 10.7 10.2

Laboratory applied field and inclination values: 100 lT, 30° 10 1.27 21.8 5.8 1.63 21.7 20 2.87 20.9 10.9 2.76 21.5 30 3.89 20.4 16.8 3.59 20.6 60 4.39 23.7 23.3 3.95 23.6 120 4.71 23.7 30.6 4.27 23.3 180 4.80 23.9 33.2 4.34 23.3 1080 4.86 23.8 35.8 4.44 23.2 4080 5.01 22.9 36.7 4.56 22.7 14400 4.98 23.8 36.2 4.53 23.0

7.3 12.1 17.4 24.1 30.6 33.5 35.7 36.3 36.5

2.10 3.26 4.04 4.37 4.60 4.71 4.77 4.88 4.81

22.1 21.0 20.6 22.8 22.4 22.8 22.7 21.5 22.4

8.8 14.5 20.5 26.0 31.8 33.7 36.3 37.2 37.1

1.25 2.11 2.56 3.16 3.53 3.60 3.78 3.80 3.88

21.3 22.1 22.4 23.4 22.9 23.5 23.2 23.8 23.5

2.9 4.7 5.9 7.8 8.5 9.1 9.5 9.5 9.6

1.36 2.02 2.41 2.99 3.24 3.34 3.45 3.52 3.54

21.8 21.8 22.0 21.8 22.7 22.9 23.5 23.4 23.1

3.4 4.4 6.2 7.9 8.8 9.1 10.0 9.7 9.7

1.77 2.26 2.58 3.24 3.52 3.67 3.84 3.83 3.92

22.1 22.8 22.5 22.6 22.8 23.0 23.1 23.9 23.5

3.9 5.2 6.2 7.5 8.5 9.2 10.2 10.0 9.9

Laboratory applied field and inclination values: 25 lT, 0° 10 0.74 2.2 5.7 0.63 20 1.72 3.0 15.0 1.11 30 1.97 1.9 18.5 1.43 60 2.22 1.9 26.9 1.85 120 2.41 2.0 31.8 1.63 180 2.36 0.7 34.0 1.23 300 2.54 0.1 36.3 1.48 1080 2.67 0.7 37.6 2.34 4080 2.70 0.7 37.7 2.45 14400 2.66 1.5 37.1 2.29

0.0 5.7 5.2 3.2 5.5 5.1 7.6 1.9 2.7 3.3

8.2 17.9 21.2 27.9 33.9 38.3 39.3 38.2 37.6 38.6

0.35 0.44 0.36 0.41 0.54 0.37 0.65 1.31 1.98 1.44

1.2 1.1 0.7 0.2 0.8 0.1 0.6 8.0 9.7 8.9

89.8 74.1 62.5 54.5 48.1 57.2 50.0 43.1 40.9 42.3

0.48 0.97 1.26 1.43 2.06 2.30 2.41 2.56 2.55 2.49

2.8 1.6 0.5 1.4 2.1 2.1 1.3 2.2 1.0 2.1

2.7 4.2 5.4 7.3 8.2 8.3 8.7 8.7 9.5 9.5

0.48 0.93 1.17 1.56 1.81 2.14 2.28 2.47 2.40 2.22

1.6 1.0 0.2 0.0 1.7 1.6 0.6 0.2 1.5 0.0

2.8 4.3 5.4 7.2 7.9 8.3 8.8 9.3 9.7 10.4

0.97 1.48 1.75 2.13 2.33 2.41 2.49 2.61 2.70 2.18

1.2 1.0 0.9 0.9 0.9 1.3 0.9 1.0 0.1 0.5

3.5 4.9 5.8 7.7 9.0 9.1 9.5 9.9 9.9 12.3

Laboratory applied field and inclination values: 50 lT, 0° 10 1.39 1.3 6.9 1.47 20 2.25 0.4 11.7 2.18 30 3.05 0.8 19.8 2.84 60 3.43 0.3 26.7 3.22 120 3.57 0.6 31.6 3.36 180 3.67 0.7 34.1 3.41 300 3.71 0.1 35.5 3.45 1080 3.74 0.2 36.9 3.46 4080 3.73 0.3 37.9 3.47 14400 3.46 9.1 38.7 3.50

0.1 0.9 1.2 0.5 0.7 1.8 0.8 0.3 0.6 6.4

8.4 12.9 20.8 27.4 31.8 33.6 35.6 36.8 38.1 39.3

1.23 2.00 2.02 2.70 2.73 2.90 3.05 2.93 2.91 3.22

3.8 1.1 3.6 0.7 5.7 3.6 2.0 2.8 5.6 1.5

14.1 18.2 26.2 30.8 35.5 36.9 37.9 40.6 41.4 42.0

0.95 1.69 2.14 2.58 2.80 2.86 2.96 3.03 3.02 3.07

1.5 1.1 1.1 0.4 0.2 0.5 1.5 1.7 0.4 0.7

2.9 4.5 5.7 6.5 7.9 8.4 8.7 9.3 9.0 9.4

1.07 1.66 2.06 2.48 2.69 2.74 2.84 2.94 2.93 2.92

0.1 1.0 0.8 0.1 0.3 0.9 0.4 0.4 0.1 0.8

3.2 4.9 6.0 7.4 8.5 8.7 9.5 9.5 9.6 9.5

1.29 1.92 2.26 2.66 2.87 2.98 3.07 3.15 3.17 3.22

0.6 0.0 0.3 0.3 0.3 0.3 0.1 0.2 0.1 0.4

3.8 5.1 6.1 7.5 8.6 9.2 9.9 10.2 10.1 9.7

Laboratory applied field and inclination values: 100 lT, 0° 4080 1.96 1.3 44.2 0.84 1.8 14400 4.45 0.1 39.2 3.16 5.5

51.2 41.6

0.64

2.3

65.3

3.30 3.25

0.6 0.5

9.5 9.5

2.93 3.11

0.2 0.5

10.1 9.5

3.51 3.50

0.1 0.3

10.1 10.0

D. Bilardello et al. / Physics of the Earth and Planetary Interiors 214 (2013) 1–13

21.9 22.2 22.2 22.3 22.3 22.5 23.9 23.8

D. Bilardello et al. / Physics of the Earth and Planetary Interiors 214 (2013) 1–13

7

Fig. 3. Data from the deposition experiments conducted in 2.0 cm diameter tubes. Results are subdivided by total magnetic moment, recorded inclination and thickness (height) of the sedimentary column. Experiments were carried out in three different applied field inclinations (IB) as indicated in each panel and in three different field intensities as indicated in color. Circles, squares and triangles are data from the different tubes; thick lines are the average results from the three tubes.

Fig. 4. Data from the deposition experiments conducted in 3.6 cm diameter tubes. Results are subdivided by total magnetic moment, recorded inclination and thickness (height) of the sedimentary column. Experiments were carried out in three different applied field inclinations (IB) as indicated in each panel and in three different field intensities as indicated in color. Circles, squares and triangles are data from the different tubes; thick lines are the average results from the three tubes.

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D. Bilardello et al. / Physics of the Earth and Planetary Interiors 214 (2013) 1–13

Table 2 Summary of the experimental and numerical results. From the data in Table 1, we calculated the mean of the last three experimental steps for each of the three tubes, and then took the average and single standard deviation (s.d.) of those three mean values. Abbreviations are: B, applied magnetic field intensity; M, magnetic moment in A m2; IR, remanent inclination (in °); IB, applied field inclination (in °). B (lT)

M ± s.d. (106)

IR ± s.d.

IR/IB

Thickness ± s.d.

2.0 cm diameter tubes IB 60° 25 50 100

2.16 ± 0.08 3.17 ± 0.12 4.33 ± 0.23

39.7 ± 2.1 48.7 ± 0.6 52.0 ± 0.5

0.7 0.8 0.9

37.9 ± 0.9 38.4 ± 0.4 37.8 ± 0.3

IB 30° 25 50 100

2.54 ± 0.09 3.55 ± 0.16 4.76 ± 0.23

15.9 ± 1.6 23.1 ± 0.5 22.9 ± 0.7

0.5 0.8 0.8

38.6 ± 1.0 37.5 ± 0.3 37.6 ± 0.4

IB 0° 25 50 100

2.52 ± 0.22 3.38 ± 0.32 –

1.1 ± 2.2 0.9 ± 2.9 –

– – –

40.4 ± 2.5 40.3 ± 2.0 –

3.6 cm diameter tubes IB 60° 25 50 100

2.28 ± 0.08 3.38 ± 0.13 4.36 ± 0.29

42.0 ± 0.8 46.5 ± 0.8 50.7 ± 0.6

0.7 0.8 0.8

11.2 ± 0.4 11.3 ± 0.5 11.8 ± 0.3

IB 30° 25 50 100

2.45 ± 0.12 3.37 ± 0.16 3.73 ± 0.20

17.6 ± 0.3 20.7 ± 0.6 23.4 ± 0.1

0.6 0.7 0.8

11.9 ± 0.3 11.4 ± 0.4 11.2 ± 0.2

IB 0° 25 50 100

2.46 ± 0.09 3.05 ± 0.13 3.27 ± 0.24

0.9 ± 0.7 0.3 ± 0.5 0.1 ± 0.5

– – –

11.3 ± 0.7 11.0 ± 0.4 11.2 ± 0.3

Experiments 3.6 diameter tubes B (lT) IR f

M

Mrel

Roll–Slip model IO f Mrel (100%)

50 (IB 60°) 50 (IB 30°) 50 (IB 0°)

3.38 3.37 3.05

0.22 0.22 0.20

48.1 20.4 00.4

46.5 20.7 00.3

0.61 0.65 –

0.64 0.64 –

0.48 0.60 0.66

gest that the magnetization acquisition process is concentration dependent. The fact that the experiments of Tauxe and Kent (1984), which had concentrations comparable to our large tubes, yield more linear trends would suggest otherwise. However, one must exercise caution since the hematite platelets present in their experiments probably behave differently than spheres, and yield different trends. Furthermore, because their experiments were conducted for shorter durations, the magnetic moments may not have been completely saturated.

on inclination shallowing recognized that sediments composed of ellipsoidal-shaped particles should experience more inclination shallowing than those composed of spheres (King, 1955; Griffiths et al., 1960; King and Rees, 1966). Like Johnson et al. (1948) and King (1955) we also observe a field dependence of the inclination error. One possibility is that this may be unique to spherical particles, for which rotation is proportional to field intensity, unlike for ellipsoids. Another explanation will be proposed in the following chapter. 5. Comparison to numerical models We now assess the different phases of particle settling and DRM acquisition numerically. We first estimate how fast particles settle, and whether the measured sediment height is consistent with the numerical simulations of settling. Sediment height is a valuable independent control because, when comparing with the evolution in magnetic moment, it confirms that a substantial fraction of beads must fall individually, rather than in flocs. The settling rate also dictates how much time the particles have to orient in the field during their fall in the column or to be randomized by Brownian motion and mutual interactions. Through numerical simulation of this process we show that the beads in the suspension are not fully oriented by the field when they reach the substrate and that their inclinations can already be shallower than the field inclination before hitting the bottom. Finally, using the model of Jezek et al. (2012), we establish that rolling and slipping on the substrate contributes significantly to inclination shallowing. Mutual magnetostatic interactions of the spheres in our experiments are negligible due to their low magnetizations and clumping plays no significant role in the shallowing process. 5.1. Settling At small Reynolds numbers, the falling velocity of spherical particles is given by the Stokes formula

v ðrÞ ¼

2 ðq  qw Þgr 2 9 l

ð1Þ

where q and qw are densities of the particle and water, r is particle radius, l is viscosity and g is gravitational acceleration. The Stokes velocity formula (Eq. (1)) should not be used for submicron particles since they undergo Brownian motion (Stacey, 1972; Collinson, 1965; Allen, 2003), as discussed in detail below. The time needed for a particle to fall from top to bottom of a water column of height hT by this velocity is

tb ðrÞ ¼ hT =v ðrÞ

ð2Þ

4.2. Recorded inclination The results from our experiments unambiguously show a field dependence on the inclination error (Fig. 5b). Like the moment data, the change in inclination error with applied field can also be globally described by a logarithmic-type function. Fig. 5c plots the recorded inclination (IR) against the applied field inclination (IB); the dashed lines correspond to different flattening factors (f) ranging between 0.5 and 0.8. The range of sediment concentrations (i.e., tube sizes) used in our study does not appear to systematically influence the amount of inclination shallowing, consistent with the findings of Blow and Hamilton (1978) and Barton et al. (1980). In fact, the variability in the trends observed in Fig. 5 is unexpected, yet must be considered inherent in the experiments since those trends were reproducible in three different tubes. Fig. 5b also plots the data of Tauxe and Kent (1984), where no field dependence of the inclination error is observed. Early work

At tb, all particles of a given radius should be settled. Fig. 6 shows tb as a function of particle radius (solid line). To take into account the concentration of a sedimenting suspension, the velocity v can be corrected by a multiplicative factor f/(t) = (1  /(t))n where /(t) is particle volume fraction and n  5 (Richardson and Zaki, 1954; Snabre et al., 2009):

v c ðr; tÞ ¼ v ðrÞf/ ðtÞ

ð3Þ

The settling times computed using the corrected velocity vc and initial concentration are shown in Fig. 6 for both 3.6 and 2.0 cm tubes (dashed and dotted lines, respectively). Concentration decreases during settling and therefore a more realistic estimate of the time at which all particles of a given radius are settled lies between the corrected curves (for t < 300 min) and Stokes’ curve. We use the time tb defined by Eq. (2) as a reference value. The measured sediment height becomes nearly constant for times

D. Bilardello et al. / Physics of the Earth and Planetary Interiors 214 (2013) 1–13

9

Fig. 5. Summary of the deposition results reported in Table 2. (A) Magnetic moment versus applied field intensity B (tube size and field inclination are specified). Results from Tauxe and Kent (1984) are plotted for comparison (TK, moments scaled proportionately to this study). (B) Inclination error (IR/IB) as a function of B for experiments performed in field inclinations (IB) of 30° and 60° (TK from Tauxe and Kent, 1984). (C) Applied field inclination (IB) versus measured field inclination (IR) for experiments carried out in 25, 50 and 100 lT. Large open and small solid symbols refer to 3.6 and 2.0 cm tubes, respectively. Dashed lines represent equal flattening factors [f = tan(IR)/tan(IB)], with f ranging from 0.5 (more shallowing) to 0.8 (less shallowing).

HS ðtÞ ¼

V S ðtÞ Sfpack

ð5Þ

where fpack is the packing ratio (the ratio of the volume of spheres to the volume they occupy). Fig. 7 compares several estimates of HS(t) obtained by numerical simulation [Eqs. (4) and (5)] with the sediment height observed experimentally. We use fpack = 0.55, which corresponds to the porosity of random loose packing, consistent with wet-packed porosities of synthetic sands with lognormal size distributions (Weltje and Alberts, 2011). We assume there is negligible compaction in the initial stages of sedimentation; initial packing would be lower if flocculation occurs. In Fig. 7, the curve computed for fpack = 0.55 (Eq. (5)) almost reaches the average mea-

Fig. 6. Settling time (tb) for particles of different radii: solid line is the curve according to Eq. (2); dashed and dotted lines correspond to the large (3.6 cm) and small (2.0 cm) tubes, respectively, after correcting for particle concentration (Richardson and Zaki, 1954).

>300 min, which according to Eq. (2) is approximately the time of settling of particles having r  1.5 lm. We can thus assume that the smallest particles never settle and remain in the column. The total volume of settled particles increases in time as an integral

V S ðtÞ ¼ S

Z 0

t

Z

v c ðr; tÞf ðr; tÞdrdt

ð4Þ

r

where S is the surface of the bottom and f(r,t) is the size distribution of the sedimenting particles. Due to different settling velocities of particles of different radii, the size distribution of the particles in the column changes over time leading to a fining upward of sediment in the deposited material. Sedimenting spheres also create pore spaces as they settle. The theoretical height of sediment in a tube is therefore

Fig. 7. Theoretical settling curves compared to experimental measurements. Experimental data are shown by solid points (blue for large and red for small tubes), white circles are the mean values. Thick black lines show how the sedimentation of individual beads should proceed according to Stokes’ velocity corrected by the instantaneous concentration and random loose packing (packing ratio fpack = 0.55). The shaded area is defined by an interval of fpack from 0.50 to 0.55. Dashed lines were computed by allowing spheres of radii r < 4 lm to fall in flocs of size 5 lm. Thick solid blue or red lines correspond to shortened floc-existence times (see text for details). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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D. Bilardello et al. / Physics of the Earth and Planetary Interiors 214 (2013) 1–13

sured heights for the 3.6 cm tubes and is visibly lower for the 2.0 cm tubes. The area between this curve and the curve for 10% lower packing (fpack = 0.50) includes the mean values, except for the first time step in the 3.6 cm tubes and the first three time steps for the 2.0 cm tubes. We also tested the possibility that particles fall in flocs that could be inherited from imperfect disaggregation of the sediment from the previous experiment and/or from clumping of particles during the fall. This may particularly occur in the 2.0 cm tubes, which possess 3.9 times higher concentrations than the 3.6 cm tubes. If we hypothesize, for example, that all particles of radii r < 4 lm fall in flocs of 5 lm radii (we use fpack = 0.55 and Eq. (1) with a corresponding change of radius and density) we obtain a curve that fits the mean heights for the initial time steps (620 min) but overshoots the points at higher time steps (dashed lines in Fig. 7). A better agreement can be attained by limiting the existence of flocs to a short interval at the beginning of the experiments. By allowing flocs to exist for 10 min, and then to settle as individual particles, we obtain a very good agreement with the average measured sedimentation heights. We also obtain a reasonably good fit when we let only one-half the number of particles of radii r < 4 lm fall in flocs for 20 min and then sediment as individual spheres. However, further decreasing the number of particles in flocs and allowing those flocs to exist over longer times (e.g., 1/5 of particles in aggregates for the first 60 min) yields poorer fits. These numerical experiments show that we cannot reject the possibility for the existence of flocs in our experiments, especially in the 2.0 cm tubes. Nevertheless, a substantial fraction of the beads must fall individually. This is supported by the fact that the accumulation of magnetic moment in the 3.6 cm tubes mimics well the height (and volume) accumulation curves. On the other hand, the moment accumulates faster than the height in the 2.0 cm tubes, which could be caused by falling flocs. This effect will be assessed below. 5.2. Magnetization in the column Rotation of spherical particles by torque from the magnetic field is expressed as

d# mV B 1 ¼ sin # ¼  sin # dt 6l sm

ð5Þ

where # is angle between the particle’s magnetic moment and the field direction, mV is the particle’s magnetization and B is the field intensity (Nagata, 1961). The characteristic aligning time (sm) lasts seconds to minutes (Nagata, 1961; Shcherbakov and Shcherbakova, 1983; Jezek and Gilder, 2006). sm corresponding to our experiments is 18, 36 and 72 s for B of 100, 50 and 25 lT, respectively. These characteristic times can be compared to the falling time through the tube (tb) and to the Brownian relaxation time, sB = 3 Vl/kT (Debye, 1929), which indicates how fast an aligned system becomes randomized in a zero field. When sB > sm, particle rotation into the magnetic field direction is significantly disturbed. The transition between Brownian and magnetic field dominated rotation occurs with beads of r  2 lm at 100 lT and r  3 lm at 25 lT (Fig. 8). Particles have enough time to be aligned by the field when tb > sm for all radii, except during the initial stages when the larger particles close to the bottom have insufficient time to become fully oriented. If we permit the presence of flocs, as indicated in the previous discussion of settling, their orientation with the field can also be roughly assessed by Eq. (5). For example, 5 lm-radii flocs with about 50% porosity will take 60 min to settle (tb). If the flocs were pre-formed due to the incomplete disaggregation of the substrate material from the previous experiment, their magnetizations would correspond to that observed experimentally, being about

Fig. 8. Comparison of the role of Brownian motion (characteristic time, sB), field alignment (characteristic time sm) and settling (tb) for the conditions in this study. Particles with radii <2 lm are influenced by Brownian motion, above 3 lm, alignment by the magnetic field prevails.

7–16% of the magnetization of beads. The characteristic aligning time sm for these flocs would be between 2 and 17 min, depending on the field strength. Before reaching the bottom, such flocs may be preferentially oriented parallel to the field direction—especially if they were inherited since the moments of the individual grains would be aligned. Due to their rugged surface, the flocs may either be immobilized when they hit the surface and preserve the preferred orientation, or they may roll as a whole. This effect would explain the faster moment accumulation and the larger variability in inclination observed in the 2.0 cm tubes. From the characteristic times of the individual beads, one expects that they are well oriented with the field direction when approaching the bottom. However, the beads will interact during their fall. Larger spheres that fall faster will collide with smaller beads below them, and both will be deflected. A larger particle of radius R may hit on average n number of smaller particles of radius r per second, given as:

n ¼ pðR þ rÞ2 ðv ðRÞ  v ðrÞÞcðrÞ

ð6Þ

Davis (1992) and Zhao and Davis (2002) studied the situation when a faster sphere falls past another in low Reynolds number fluids. They considered two cases: one of rigid body rotation where spheres rotate together before splitting, and a second of mutual rolling with possible slipping after the maximum friction force is reached. The second, roll-slip, model provided a better fit to their experimental data and will therefore be used here. In this model, the larger (L) and smaller (S) particles rotate by different angular velocities whose ratio is

xL b1 þ c1 minð1; cot h= cot hS Þ ¼ xS b2 þ c2 minð1; cot h= cot hS Þ

ð7Þ

where b1, b2, c1, and c2 are given by means of the two-sphere mobility functions of Jeffrey and Onishi (1984), and h is the angle between the line of centers of the two spheres and the vertical, i.e., the instantaneous slope. hS is a threshold angle. For angles h < hS, spheres roll without slipping; slipping occurs when h P hS. Zhao and Davis (2002) found that realistic values of hS are 15–30°, which is also supported by experimental and theoretical evaluation of spheres rolling down an inclined plane (Smart et al., 1993; Galvin et al., 2001; Zhao et al., 2002). A numerical model of the magnetization in the column was constructed as follows. We consider a multi-particle system composed of particles of radii r = 3, . . . , 11 lm (particles smaller than 3 lm are not considered as they are influenced by Brownian motion). The initial particle size distribution corresponds to our

D. Bilardello et al. / Physics of the Earth and Planetary Interiors 214 (2013) 1–13

experiments, and then the size distribution evolves in time according to Eq. (4). At each time step, particles are oriented by the magnetic field (Eq. (5)) and disoriented by collision. The number of hits is given by Eq. (6). Each interaction causes both particles to rotate around a horizontal axes. The larger particle can hit the smaller one at any position on its upper hemisphere (determined randomly) that in turn defines the angle h, the azimuth of the rotation axis, and the amount of rotation (Eq. (7)). We use hS = 25° and the ratio of slipping to rolling 0.5. This means that for h < 25° (when a larger particle hits a small one close to its top), particles fully rotate, and for h > 25° (when a larger particle hits a small one more to its side), particles half rotate and half slip. This model provides an estimate for the magnetization of the suspension in the column before reaching the substrate. Results of the simulations (Fig. 9) indicate that (1) the particles are not fully oriented with the magnetic field, (2) the inclination is shallower than expected, and (3) there exists a significant field dependency on the magnetization and inclination. Because equation (6) also accounts for concentration, there is a clear difference between the two tube-sizes, with the 2.0 cm tubes showing lower moments. It is probable that mutual interactions in the column, especially in the 2.0 cm tubes with higher concentrations, leads to rolling and slipping processes away from the ideal case described by the above model. But even if rolling and slipping only partially occurs, the results from the 2.0 cm tubes will still be more influenced, yielding lower net moments and shallower inclinations when arriving at

11

the substrate. This in turn should contribute to lower moments and shallower inclinations upon final deposition. Note that, because the mutual rotation of colliding spheres proceeds around horizontal rotation axes, the nature of inclination shallowing in the column is similar to the Griffiths et al. (1960) model of inclination shallowing. 5.3. Rolling and slipping on the substrate King (1955) assumed that spherical particles deposited on a substrate will correctly record the field inclination, whereas Griffiths et al. (1960) showed that spherical particles hitting the bottom roll equally in all directions, thus producing inclination shallowing. However, the model of Griffiths et al. (1960) is limited by not-knowing a representative value of the angle of rotation – this ambiguity persists until today; moreover, they considered only populations of equi-dimensional spheres. To overcome these pitfalls we constructed an algorithm that treats random sedimentation with rolling and slipping spherical particles (Jezek et al., 2012). Here we briefly describe the model and its implications to our experiments. We generate spherical particles in random x, y and z coordinates above a substrate and allow them to fall until they hit previously deposited spheres (Fig. 10). Depending on the geometry of the spheres on the substrate and where the descending particle encounters the substrate, several scenarios are possible concerning

Fig. 9. Results from numerical simulations of the suspension above the substrate. Thick lines are for 3.6 cm tubes, thin lines for 2.0 cm tubes. Green, blue and red correspond to fields 25, 50 and 100 lT, respectively. Field inclination in this case was 60°. Mr is the magnetization relative to the saturation value; IO is the observed inclination.

Fig. 10. Random sedimentation of spherical particles of variable sizes.

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how/if the particle will roll according to the geometry of the bottom layer. The end result occurs when the sedimenting sphere arrives at a stable position (stops moving), supported by at least three stationary spheres, and becomes embedded into the substrate. By repeatedly solving the equations of mutual contact of two or more spheres, we track the sphere’s path and record the magnetic vector upon cessation of motion. This is done for tens of thousands of particles. Not only do the particles roll, but they can also slide. As described above we define slipping by means of the threshold angle hS and the ratio of slipping to rolling when the angle hS is exceeded (see Jezek et al. (2012) for a detailed description). In the simulations we used hS = 25° and the ratio of slipping to rolling equals 0.5, as in Section 5.2. Repeated simulations of this model show that a characteristic value of the inclination shallowing factor (f) is on the order of 0.6 (simulations with equal-sized spheres produced a standard deviation of 0.02). Interestingly, f  0.6 also holds valid for spherical populations with non-equal radii, including uniform (i.e., same number of spheres of given r) and exponential size-distributions. The latter can be taken as an end member for our experiments. Note that the model is sequential in that each sphere stops rolling before another one falls. There is no influence of the sphere’s momentum. A falling or depositing sphere on the substrate cannot displace another sphere, cause avalanches, etc. In the lower part of Table 2 we compare the average measured inclination, IR, and relative moment (Mrel) in the large tubes for IB = 60° with those predicted by the roll-slip model. In these simulations all beads were perfectly oriented parallel to the field before reaching the substrate and then subjected to rolling and slipping. Therefore the results reflect the separate role of rolling and sliding on the substrate. The inclinations are well predicted, but the model yields higher relative moments than the experimental values. However, it was shown above that particles are not fully aligned with the field when they reach the bottom, so the moment given in Table 2 must be lowered accordingly, which will be different for small and large tubes. Although the moment and inclination of the suspension above the substrate are influenced more in the small tubes than in the large ones (Fig. 9), proportional to particle concentration, the situation is reversed on the substrate—the higher concentration of the small tubes restricts rolling and sliding more than in the large tubes. Both processes are complementary (counteracting), which can partly explain small differences observed between results from the large and small tubes. To summarize the numerical models, we find that one factor controlling the observed field dependence on both moment and inclination originates while the particles are in the water column due to mutual interaction. An additional process is the reorientation by magnetic torque of particles lying on the surface before they are buried by newly falling particles. During slipping, the spheres can reorient to the field direction in response to the field. Furthermore, the proportion of particles controlled by Brownian motion is also field-dependent (Fig. 8). Finally, stable recorded moments and inclinations over longer time scales show that realignment of particles in the substrate seems to be inhibited or very restricted by grain–grain contact; pDRM effects, if present, are hardly discernible.

6. Conclusions Our experimental set up was designed to examine the remanent magnetization acquisition process during sedimentation of solely spherical particles in the absence of flocculation. These experiments and numerical models represent an end-member of a myriad of potential natural scenarios, yet they highlight how spherical particles, in particular, should behave in the natural environment.

The main results can be summarized as follows: (1) For two different diameters of settling tubes, the detrital remanent magnetization increases then plateaus at a nearly constant level as the settling time increases. (2) With few exceptions, the measured sediment inclination is shallower than the ambient field inclination from the outset of the experiment, i.e. at the shortest settling times. No discernible change in inclination is observed as the settling time increases. Inclination shallowing is similar to what is observed for natural sediments, with a flattening factor of about f = 0.6; e.g., sediments composed entirely of spherical particles can produce inclination shallowing, on the order observed in nature. (3) The amount of inclination shallowing generally follows the King (1955) tangent-tangent relationship with respect to the magnetic field inclination. Moreover, the degree of inclination shallowing depends on the field strength, with less shallowing for stronger magnetic fields. Through numerical modeling, we improved upon the model of Griffiths et al. (1960) by calculating the amount of rotation produced by mutual interaction of spherical particles in the column and their rolling on a surface created by randomly falling particles. A roll/slip model (Zhao and Davis, 2002) of particle interaction in the column and on the sediment–water interface accounts for lower moments and inclinations than otherwise predicted in the absence of particle interaction. The importance of rolling is damped as field intensity increases, which in turn leads to less inclination shallowing and heightened magnetization. It is possible that the flocculation process restricts rolling, hence limits the field dependency on inclination shallowing. Compaction ill explains the relationship between volume of accumulated material and acquisition of remanence in our experiments. On the other hand, the roll/ slip formulation predicts a difference in moment acquisition and inclination shallowing between the two tube diameters, as the concentration and likelihood to collide is higher in tubes with smaller diameters. However, the concentration in the small tubes may exceed the applicable limit of the roll/slip model as multiple interactions restrict rotational freedom. If true, experiments using lower concentrations (e.g., 3 g of sediment in the 3.6 cm diameter tubes) should find higher (relative) moments and lower amounts of inclination shallowing. Future deposition experiments should monitor accumulation rates and use multiple tubes to assess data reproducibility. Temperatures should be held constant during the entire span of the experiments. The relative contribution of different factors or contributions from the processes involved (rolling, flocculating, hitting, etc.) could be determined experimentally by varying the concentration and the size-distribution in future experiments. Experiments employing strictly oblate particles may become feasible in the near future with the emergence of techniques to grow hematite grains with narrow size distributions and provide an alternative end-member to the inclination shallowing problem. Acknowledgements Funding from the national science agencies of Germany and the Czech Republic supported this project (Grants DFG GI712/3-1, GACR 205/09/J028 and MSM0021620855). Jessica Till kindly performed the FORC measurements at the Institute for Rock Magnetism. We thank Ramon Egli and Michael Winklhofer for stimulating discussions. We also thank Mark J. Dekkers and an anonymous reviewer for comments that improved the manuscript. References Allen, T., 2003. Powder Sampling and Particle Size Determination. Elsevier (600pp). Arason, P., Levi, S., 1990. Compaction and inclination shallowing in deep sea sediments from the Pacific Ocean. J. Geophys. Res. 95, 4501–4510.

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