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九州大学大学院理学研究院紀要

M E M O I R S
OF THEFACULTYOFSCIENCESKYUSHU UNIVERSITY

Series D Earth and Planetary Sciences
VOLUME XXXIIINo. 1

FUKUOKA, JAPAN
December, 2013

Contents

Tasuku Akagi:
Revision of the Dissolution Kinetics of Aggregated Settling Particles ……

Editorial Board
Hiroyoshi Sano, Chief Editor
Jun-ichiro Ishibashi
Syoichi Shimoyama

All communications relating to this Memoirs should be
addressed to the Dean of the Faculty of Sciences, Kyushu University
Hakozaki, Fukuoka 812-8581, JAPAN1

Mem. Fac. Sci., Kyushu Univ., Ser. D, Earth ＆ Planet. Sci., Vol. XXXⅢ, No. 1,pp. 1-5, December 00, 2013

Revision of the Dissolution Kinetics of Aggregated Settling Particles
Tasuku Akagi

Abstract
The dissolution kinetics theory of settling particles, which was developed to express the elemental
concentration variation as a function of diatom productivity, is revised to allow further consideration on thepresence of voids. In this revision, the number of diatom frustules which form an aggregate is introduced asa basic variable to secure the rigidity of mathematical consideration, (unlike the previous paper, whereproductivity is employed as one of the basic variations - Akagi et al., 2011). No exchange of elements is
assumed between seawater surrounding the aggregates and seawater in the voids of the aggregates. It isshown that the presence of voids reduces the settling velocity, but does not affect the dissolution/alteration
behavior of settling particles as long as the particles are collected at a certain depth.
Keywords: settling particles, aggregate, dissolution theory, voids

1. Introduction
Interaction between particles and solution is one of the most important processes to control the distribution
and behavior of dissolved matter in the oceans. This interaction includes chemically describable reactions such as
dissolution, desorption, adsorption, etc. However, independent movements of particles against the medium makea seawater column a complex system. I believe that this is one of the reasons for rather inconsistent compositional
data of the settling particles (Archer, 2006), which have hampered a clear understanding of the interaction.
Akagi et al. (2011) observed that the composition of settling particles always exhibits a hyperbolic relationshipagainst opal mass flux (P) and proposed the dissolution kinetics of diatom opal, as expressed by eq. (1).
(1)
where C i , D and k d,i are concentration of the element (i) in particles, depth the particles are sampled at, and adissolution constant of i , respectively. C io indicates the concentration of i, before dissolution. I believe that thedissolution kinetics is the first successful consistent interpretation of the compositional variation of settlingparticles. In the kinetics diatom frustules aggregate to form larger particles, which transfer through a watercolumn with a smaller length of time and a lesser proportion of exposed surface to contact with seawater. As aresult of decreasing time and surface contact with seawater, the aggregates tend to retain their originalcomposition with a lesser extent of dissolution/alteration. Akagi et al. (2011) then concluded that the asymptotic
values of the hyperbolic relationships correspond to the original values of frustules (Cio in eq (1)). This has later
been verified by the diatom incorporation theory (Akagi, 2013). In the model the settling velocity and the extentof dissolution/alteration are related through the size of aggregates. However, the model does not explicitly include
the number of frustules in aggregates in the derivation of the model equations, rendering further consideration
rather difficult.
Naturally observed marine snows show variation in density (Alldredge and Gotschalk, 1988), and voids ofvarying proportion may be present in real aggregates. It is likely that the aggregates gradually lose voids andManuscript received on 25 September 2013; accepted on 6 November 2013
Department of Earth and Planetary Sciences, Faculty of Sciences, 33 Kyushu University, 6-10-1 Hakozaki, Higashi-ku,Fukuoka 812-8581, Japan

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Tasuku Akagi

become denser during settlement to increase settling velocities (Berelson, 2002). In this paper, the model ismodified to incorporate the number of frustules and the effect of presence of voids on the dissolution kinetics ofaggregates is theoretically considered.

2. Dissolution Kinetics Theory of a Single Aggregated Particle
Let m i be mass of an element, i, in a single particle. Following the dissolution kinetics (Akagi et al., 2011),
dissolution, dm i , is proportional to both contact time dt and surface area:(2)
where k i , C i and S are the proportionality constant, concentration of the element in the particle, and surface area,respectively. Adsorption reaction may follow a slightly different equation, but nonetheless similar to eq. (2) withthe opposite sign of k i and general term of dissolution/alteration can be applied to refer to the equation. The lengthof time the particle is in contact with seawater is proportional to the distance that the particle travels, D, and thereciprocal of the settling velocity, v. Therefore,(3)
Here it is shown how the three parameters, C i , S and v, are expressed as a function of the volume proportion offrustules, f , in an aggregated particle consisting of a number of diatom frustules, n (see Fig. 1). Each frustule hasa unit volume of V p and density r p . The volume of the particle, V, is expressed by(4)Assuming the particle is spherical, the radius, r, and surface area, S, of the particles are(5)
and(6)
respectively.

Fig. 1 Schematic diagram of a settling aggregated particle with voids of an aqueous medium.

Revision of the Dissolution Kinetics of Aggregated Settling Particles

3The volume proportion of voids in the particle is 1-f. If the voids are filled with seawater of density r w , thedensity of the particle, r, is(7)
According to Stokes’ law, settling velocity of the particles is(8)where k is a proportionality constant.
The concentration C i of the particles is also diluted by the voids, i.e.,(9)
Putting eqs. (6), (8) and (9) into eq. (3), we have(10)

It should be noted that the proportion of frustules in the aggregate, f, and surface area, S, are both eliminated ineq. (10).
Since the mass of the aggregate is nVp r p , the concentration of the element after a certain time is expressed asppp

(11)

p

where superscript o indicates the initial values. When the change is small, the equation can be approximated to(12)pTherefore,(13)K is an overall proportionality constant,(14)
ppEqs. (1) and (13) are identical to each other if n is proportional to P 2. The number of frustules to aggregate, n, is afunction of frustule density, but it should be noted that n is not a function of productivity, P. (Note P is expressedby mass of product per unit area per unit time and is different from the density in dimension.) It is often the casethat the number of frustules to aggregate, n, tend to increase with an increase of P. This rather vague remark on
n-P linkage is enough to explain the observed hyperbolic relationship between the concentration of elements insettling particles and productivity (Akagi et al., 2011). This also provides the rationale that one can obtain theoriginal concentration of frustules when P is great enough (Akagi et al., 2011).

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Tasuku Akagi

3. Results and Discussion
3.1. Influence of voids in the dissolution of elements in aggregatesIn eq. (13) f is eliminated. It is thus concluded that, as long as settling particles are sampled at a fixed depth,
the extent of dissolution is not affected by the change in the proportion of voids. In the model no exchange ofelements is assumed between seawater surrounding the aggregates and seawater in the voids of the aggregates. If
aggregates are packed enough, this assumption may be considered to be valid.
3.2 Influence of voids in settling velocity of aggregates

Eq. (8) indicates that the settling velocity is proportional to f 1/3. Berelson (2002) observed the increase in settling
velocities of particles with increasing depth. This consideration suggests that the decrease of a void volume withincreasing depth may be one of the reasons.

4. Conclusion
The properties of settling particles theoretically derived by this study are summarized in Fig. 2. The presenceof voids in settling particles increases particle size and the density of the particles becomes closer to the medium,compared with void-free settling particles. The settling velocity becomes lower as the results of both effects.
Because the settling particles are normally collected using a sediment trap deployed at a certain depth, it is foundthat the dissolution/alteration of the particles during the settlement of a water column is not affected by thepresence of voids and the same equation used in void-free particles expresses the dissolution of elements in awater column. The present model may show that one of the most important factors to determine dissolution/alteration of aggregates is the number of frustules to form aggregates, but not the presence of voids. The presenceof voids allows some less-altered highly-aggregated particles to settle at the same rate as highly-altered lessaggregated particles.

Fig. 2 Theoretical relationships of volume, surface area, particle size, density, settling velocity, dissolution/alteration extent of asingle aggregated particle v.s. the presence of voids (1-f ). Values are relative to the corresponding ones of the void-freeaggregated particle ( f =1). In the calculation r p is set to 2 r w .

Revision of the Dissolution Kinetics of Aggregated Settling Particles

5Acknowledgement: I thank Dr Yusuke Okazaki for his comments on the manuscript. This research wassupported by MEXT/JSPS Grant Number 23310008.

References
Akagi, T., Fu F.-F., Hongo, Y. & Takahashi, K. (2011) Composition of Rare Earth Elements in Settling Particles
Collected in the Highly Productive North Pacific Ocean and Bering Sea: Implications for Siliceous-Matter
Dissolution Kinetics and Formation of Two REE-Enriched Phases. Geochim. Cosmochim. Acta, 75, 48574876.
Akagi, T. (2013) Rare earth element (REE)-silicic acid complexes in seawater to explain the incorporation ofREEs in opal and the “leftover” REEs in surface water: new interpretation of dissolved REE distribution
profiles. Geochim. Cosmochim. Acta, 113, 174-192.
Alldredge A.L. and Gotschalk C. (1988) In situ settling behavior of marine snow. Limnol. Oceanogr., 33, 339-351.
Anderson R.F. (2006) Chemical tracers of particle transport. In H. Elderfield(ed.) Treatise on Geochemistry. Vol. 6
The Oceans and Marine Geochemistry. Elsevier, 247-273.
Archer D. (2006) Biological fluxes in the ocean and atmospheric pCO 2 . In H. Elderfield(ed.) Treatise onGeochemistry. Vol. 6 The Oceans and Marine Geochemistry. Elsevier, 275–291.
Berelson W. M. (2002) Particle settling rates increase with depth in the ocean. Deep-Sea Res. II, 49, 237–251.

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Series D, Earth and Planetary Sciences
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