P Jacquet

Non Unitary Random Walks

Keywords: random walk, black hole, unitarity, dark energy, continued fractions, asymptotic analysis marche aléatoire, trou noir, unitarité, énergie sombre, fractions continues, analyse asymptotique

Motivated by the recent refutation of information loss paradox in black hole by Hawking, we investigate the new concept of non unitary random walks. In a non unitary random walk, we consider that the state 0, called the black hole, has a probability weight that decays exponentially in e -λt for some λ > 0. This decaying probabilities affect the probability weight of the other states, so that the the apparent transition probabilities are affected by a repulsion factor that depends on the factors λ and black hole lifetime t. If λ is large enough, then the resulting transition probabilities correspond to a neutral random walk. We generalize to non unitary gravitational walks where the transition probabilities are function of the distance to the black hole. We show the surprising result that the black hole remains attractive below a certain distance and becomes repulsive with an exactly reversed random walk beyond this distance. This effect has interesting analogy with so-called dark energy effect in astrophysics.

Introduction and motivation

A black hole is a massive stellar object that absorb everything trapped by its gravitational field, including light. In 1976, Stephen Hawking [START_REF] Hawking | Breakdown of Predictability in Gravitational Collapse[END_REF], the famous astrophysicist, discovered that the black holes are not absolutely black and eternal. Indeed black holes evaporate via Hawking radiation and eventually disappear. The problem is in the fact that Hawking radiation does not carry any information and therefore the information contained in the objects absorbed by the black hole during its lifetime simply disappear. This information loss paradox leads to fundamental question in theoretical physics since it apparently violates the principle of unitarity in quantum physics.

In 2006 Hawking [START_REF] Hawking | Information Loss in Black Holes[END_REF] refuted its argument about the information loss paradox. His refutation is still the object of controversy but established the foundation of non unitary Markov system. Basically Hawking stated that if black holes would have a non unitary effect, then the impact of this non unitary effect on any black hole history would vanish exponentially with time and therefore only the effect in unitary metric would remain at infinity. In other words a black hole tends almost surely to be unitary and to release its information without loss during its lifetime.

The motivation of this paper is to extend the analysis of non unitary effect on a system involving a black hole. We describe a simple model of non unitary effect as a probability weight that decays exponentially with time. What is interesting is the effect of the unitary loss on a halo of matter that surround a black hole. The matter acts like a gas where particles collide at random times like in a random walk. In physics such random walk problems are investigated via finite element simulation with particles traveling between finite boxes of space. Each box is a state in the random walk. In a non unitary random walk, we consider that the state 0, called the black hole state has a probability weight that decays exponentially in e -λt for some λ > 0.

We will show that such non unitary random walk shows a behavior which is very sensitive to its tuning parameters, and thus the simulation must be done in the most realistic situation. The number of states needed to simulate a random walk in a galactic halo may exceed 10 20 , assuming spherical symmetry in order to consider only distinct distances to the central black hole. This makes the simulation very hard to perform. Furthermore the impact of the unitarity effect stands in the future of the particle, namely in the quantity of probability weight it will lose after eventually ingested by the black hole. Therefore the simulation must compute the probability weight of all potential trajectories. In a (classic) unitary universe, all these trajectories sums to one. But in a non unitary universe this is not the case, and computation must involve all possible trajectory within the whole life time of the black hole. A super-massive black hole, like the black hole in the center of our galaxy, has a lifetime of order 10 90 years. Such simulation is rather impossible to make: either we have to iterate on vector of 10 20 coefficients over 10 90 steps, or we iterate during log 2 10 90 steps, on the successive squares of the non unitary matrix 10 20 × 10 20 . The object of this paper is to analytically guess the results of such simulations by investigating the asymptotics of non unitary random walks.

We show the surprising result that the black hole remains attractive below a certain distance and becomes repulsive or at least neutral beyond this distance.

This effect shows an interesting analogy with so-called dark energy effect in astrophysics.

Non Unitary effect and the Schrödinger's rabbit

In a (honest) unitary universe all probabilities sum to one. If you take a rabbit at time t = 0 then one month later the rabbit will be either alive with probability α > 0 or dead with probability β > 0 and α + β = 1. But in quantum physics the sum of probabilities is equal to the integral |Ψ| 2 of the wave function Ψ of the universe. Therefore the sum of probabilities being equal to one is not only a mathematical statement but also a physical statement. Unitarity principle states that for all time t: |Ψ| 2 = 1. But in a non unitary universe we may have |Ψ| 2 = 1. The loss of information implies a non unitary universe. Hawking mentions metrics pertinent for black holes (so-called anti-deSitter metrics) where |Ψ| 2 = e -λt for some λ > 0. In other word let assume a rabbit which at time t = 0 is either in state s 0 inside a black hole with a non unitary metric with probability α or in state s 1 outside a black hole in an unitary metric. If blackhole lifetime is t then at time t the probability weight of state s 0 will be r 0 (t) = e -λt while the probability weight of state s 1 will remain at r 1 (t) = 1. Consequently the sum of probabilitities of the rabbit equals αr 0 (t) + βr 1 (t) = 1 and when t increases the apparent probability of state s 0 defined as αr0(t) αr0(t)+βr1(t) → 0. In other words the rabbit never falls in non unitary metric and stays in unitary metric. This is equivalent as if an investor has one euro at time t = 0, and put α cents in a black hole market, and keep β cents in his pocket. If the black hole market looses value at rate -λ, then at the end of day the investor will have most of his remaining money in his pocket.

In fact Hawking point is not as trivial as exposed above: he does not consider a rabbit, but the whole black hole and considers that its quantum state is a superposition of unitary and non unitary metrics. He shows that on the path integral over the black hole lifetime, the contributions of non unitary metric vanish exponentially, so that only the unitary metric contributions remain.

In the following we are not considering state superposition, we assume that either the rabbit is inside the black hole (state s 0 ) or far away outside the black hole (state s 1 ) so that the state functions don't overlap. In this hypotheses we treat the problem in a classical way. We can see the rabbit as a classical equivalent of Schrödinger cat in an non unitary universe. The Schrödinger cat can be in the superposition of two states s 0 and s 1 (for instance dead or alive). We introduce the Schrödinger rabbit which is either in state s 0 or in state s 1 , but probability weight of state s 0 decays exponentially with time.

We shall notice that black hole typical lifetime t can be extremely large. In particular massive black holes at center of galaxies with mass of about 10 4 solar masses, would evaporate in a time that exceeds 10 80 years [START_REF] Traschen | An Introduction to Black Hole Evaporation[END_REF] greatly exceeding the current age of our universe (10 10 years)! 3 Modeling non unitary systems

Non unitary Markov processes

We still consider a two state process with state s 0 and a state s 1 . State s 0 is an absorbing state that mimics a black hole and during the black hole lifetime the probability weight decays with rate -λ. We consider that the time is discretized INRIA and during the black hole lifetime, at each time unit the rabbit has a probability q to fall in the black hole and probability p. When the rabbit is in the black hole it stays inside until the end of the black hole lifetime.

Or in other words, an investor has one euro in its pocket at time t = 0 and he has a contract with its black hole bank to move every month a fraction q of his pocket money in the black hole market.

This system mimics the effect of a black hole which is permanently attractive on any object in its vicinity. We will see that if the black hole exerts an attraction which is more effective than its unitary decay, then the previous result is highly modified.

We still have r 0 (t) = ρ t , but about r 1 (t), the transition from lifetime t and lifetime t -1 follows the identity r 1 (t) = qr 0 (t -1) + pr 1 (t -1) which makes r 1 (t) < 1. Notice that the previous identity is time forward, since the black hole lifetime decreases when the time goes forward. We call such system a non unitary Markov process with transition matrix: R = ρ q 0 p .

We call ρ the one-step unitary decay. The matrix R is not unitary in the Markovian sense, since vector [START_REF] Gerl | Continued fraction methods for random walks on N and on trees[END_REF][START_REF] Gerl | Continued fraction methods for random walks on N and on trees[END_REF] is not an eigenvector and 1 is not an eigenvalue:

[1, 1]R = [1, 1] .
The eigenvalues of matrix R are ρ and p. We expect to have different behaviour when p > ρ or when ρ > p. The objective is to estimate the apparent attraction of the black hole, or more precisely, the apparent repulsion p(t) = p r1(t-1) r1(t) . We have

[r 0 (t), r 1 (t)] = [1, 1]R t .
We prove the following very simple theorem whose purpose is to introduce our methodology Theorem 1. In the two-state model, when ρ > p the black hole is apparently attractive, when ρ ≤ p, the black hole is apparently repulsive when t → ∞.

Proof. From the fact that r 1 (t) = q ρ t -p t ρ-p + p t we get p(t) = p ρ + O(( p ρ ) t ) when ρ > p. In this case the attraction is stronger than the non unitary effect and the rabbit eventually falls in the black hole which exert an apparent attraction 1 -p(t) still stronger than the apparent repulsion p(t). In the opposite case, when ρ < p, we have p(t) = 1 + O(( ρ p ) t ), the non unitary effect is stronger than the attraction and the black hole is apparently fully repulsive. When p = ρ, we get r 1 (t) = tqp t-1 + p t and p(t) = 1 + O( 1 t ).

Non unitary random walks

In this section we investigate the system where there are several states: an absorbing black hole state s 0 and a infinite sequence of states s 1 , s 2 , . . . , s n , . . .. If the rabbit is in state s 0 its stay there for ever, or at least for the remaining black hole lifetime, if the rabbit is in state s n , for n > 0, then at the time unit it has probability q to shift down to state s n-1 , and probability p to shift up to state s n+1 . We denote r n (t) the sum of probabilities when the rabbit is on state s n when the black hole has a remaining lifetime t. We denote K n (u) = t≥0 r n (t)u t .

Let T n the time at which the rabbit hits the black hole starting at state s n at time 0. We have r n (t) = θ≤t P (T n = θ)ρ t-θ + P (T n > t), and thus

K n (u) = F n (u) 1 1 -ρu + (1 -F n (u)) 1 1 -u RR n°0001
with F n (u) the probability generating function of T n . F n (u) satisfies F n (u) = quF n-1 (u) + puF n+1 (u), and therefore F n (u) = (F (u)) n , with 1 = qu F (u) + puF (u), which leads to

F (u) = 1 -1 -4pqu 2 2up .
Our aim is to evaluate the apparent repulsion pn (t) = p rn+1(t-1) rn(t)

. By Cauchy we have

r n (t) = 1 2iπ K n (u) du u n+1 .
The generating function K n (u) has two main set of singularities, one at u = 1 ρ and another one at ±u(p) with u(p) = 1 2 √ pq . We expect a change in behavior when one has smaller modulus than the other one. Notice that u = 1 is not a singularity since 1 -F n (1) = 0 and by Lhospital rule, K n (u) can be analytically extended beyond u = 1.

Case ρ > 2 √ pq

Theorem 2. When ρ > 2 √ pq, the black hole is apparently attractive for all states s n with n > 1, when t → ∞.

Proof. In this case u = 1 ρ is the dominant singularity in K n (u) and we get the estimate

r n (t) = (F ( 1 ρ )) n ρ t + O((u(p) -t ) , and pn (t) = p ρ F ( 1 ρ ) + O((ρu(p)) -t ) .
We notice that pn(t)

p > 1 since 1 = q ρF ( 1 ρ ) + p ρ F ( 1 ρ ) > p ρ F ( 1 ρ ).

Case ρ < 2 √ pq

Theorem 3. When ρ < 2 √ pq, the black hole is apparently repulsive for all state n with n > 1, when t → ∞ and n = o( √ t).

Proof.

In this case u = ± 1 2 √ pq are the dominant singularities in K n (u). Let u(p) = 1 2 √ pq . When u = ±(u(p) + δu) we have F n (u) = 1 (2up) n 1 -2n(pq) 1 4 √ δu + O(n 2 δu) Bounded values of n Applying Flajolet Odlyzko [3] asymptotic result, namely r n (t) = ρ n+t (2p) n + √ π n t 3 2
(pq)

1 4 (2p) n (u(p)) n+t 1 1-ρu(p) -1 1-u(p) + (-1) n+t 1+ρu(p) ) -(-1) n+t 1+u(p) + O( n t )
Notice the oscillation on the odd and even values of n + t which are due to the model artefact that the random walk only affect the closest neighbor states of the current state. These oscillation does not affect the values of the apparent repulsion pn (t) for n > 1 which satisfies:

pn (t) = 1 2 n + 1 n 1 + O( 1 t ) + O( n t )
INRIA which converge to 1 2 as soon as t → ∞ with n t → 0. In other words the apparent random random walk tends to be neutral with attraction balancing the repulsion.

For

n = 1, since r 0 (t) = ρ t ≪ r 1 (t) = O( 1 u ( p) t )
we have p1 (t) → 1: the black hole is apparently 100% repulsive on the last state before, and the random walk on the remaining states is apparently neutral. In other word the random walk is apparently neutral and reflective on the last state before the black hole.

Unbounded values of n If we consider unbounded value of n, but with n = o( √ t), we need a more involved analysis shows that since

r n (t) = 1 2iπ(2p) n 1 -1 -4pqu 2 n g ρ (u) du u n+t+1 with g ρ (u) = 1 1-ρu -1 1-u .

by deformation of the integral loop we get for any arbitrary

z < 1 ρ r n (t) = I n (t, z) + J n (t, z) + O( 1 (2p) n z n+t ) with I n (t, z) = 1 2iπ(2p) n z u(p) 1 + i 4pqu 2 -1 n -1 -i 4pqu 2 -1 n g ρ (u) du u n+t+1 (1) J n (t, z) = -1 2iπ(2p) n -u(p) -z 1 + i 4pqu 2 -1 n -1 -i 4pqu 2 -1 n g ρ (u) du u n+t+1 (2)
With change of variable u = (1 + v n+t )u(p), we get

u n+t+1 = 1 + O( v 2 n + t + 1 ) exp(v) (3) 1 + i 4pqu 2 -1 n = 1 + O( vn n + t ) exp(i n √ n + t + 1 √ 2v) (4) 1 -i 4pqu 2 -1 n = 1 + O( vn n + t ) exp(-i n √ n + t + 1 √ 2v) (5) g ρ (u) = 1 + O( v n + t ) g ρ (u(p)) (6)
and consequently

I n (t, z) = gρ(u(p)) (n+t)π(2p) n u(p) n+t+1 (z-u(p)) n+t+1 u(p) 0 sin( n √ n+t √ 2v)e -v dv + O( n n+t ) gρ(u(p)) (n+t)π(2p) n u(p) n+t+1 π 2 n √ n+t exp(-1 2 n 2 n+t ) + O( n n+t )
Similarly with the change of variable u = -(1

+ v n+t )u(p) we get J n (t, z) = g ρ (-u(p))(-1) n+t+1 (n + t)π(2p) n u(p) n+t+1 π 2 n √ n + t exp(- 1 2 n 2 n + t ) + O( n n + t )
which validate the asymptotics obtained via Flajolet-Odlyzko method to unbounded values of n, provided that n = o( √ t).

Figure 1: actual potential (dashed) and apparent asymptotic potential (plain) for p = 0.2 for ρ = 0.9 (left) and ρ = 0.81 both greater than 2 √ pq = 0.8.

Random walk potential

Let V n = n 1 2 log q p that we call the potential of state s n . Clearly the state s 0 is the state with the lowest potential. Notice that the analysis does not change if we add to the potential an arbitrary constant.

Noticing that 1-pn(t) pn(t) = q p rn+1(t-1)

rn-1(t-1) . The apparent potential Ṽn (t

) = V n + 1 2 log r n+1 (t -1) + 1 2 log r n (t -1) = i≤n 1 2 log 1-pi(t) pi(t)
. It comes from the previous analysis that when ρ > u(p) we have an apparent potential Ṽn (t) tends to be equal to n 1 2 log 1- p p , modulo an arbitrary constant term, with p = p ρ F ( 1 ρ ), and in this case state s 0 is the state with the lowest potential. The black hole is still attractive.

When ρ < u(p), the apparent potential tends to be flat, excepted for a peak on the black hole state.

Figure 1and 2 show the asymptotic apparent potentials when ρ is above or below 2

√ pq. Notice that as expected the apparent repulsion does not change very much as soon as ρ < 2 √ pq.

Non unitary concave walk

We call a concave walk a random walk where the transition pair of probabilities p n , q n depends on the state and p n increases with n with lim p n ≤ 1 2 when n → ∞. For example p n = 1 2 -β n 2 if one wants to simulate the random walk of a particle around a gravitationally heavy object such as a stellar body. Or p n = 1 2 -β n if one wants to simulate the random walk of a particle in a gravitationally heavy medium of density decreasing in the inverse of distance to the center (state s 0 ). A gravitational walk can be used to simulate the behavior of a particle in a gas where each collision with another particle give a random momentum.

Similarly as with uniform walk we define the actualpotential of the random walk as

V n = j=n j=1 1 2 log( q j p j ) .
INRIA Let G i (u) be the probability generating function of the time to get down from state s i to state s i-1 . We have

F n (u) = i=n i=1 G i (u)
We have the recursion

G n (u) = q n uF n-1 (u) + p n uF n+1 (u) , or G n (u) = q n u + p n uG n+1 (u)G n (u) .
We therefore get the recursion

G n (u) = q n u 1 -p n uG n+1 (u) , (7)
which resolves into the continuous fraction as noticed by [ref]

G n (u) = q n u 1 -pnqn+1u 2 1- p n+1 q n+2 u 2 1-••• .
We notice that for all n, function G n (u) is odd:

G n (-u) = -G n (u). Let us denote F (u, p) = 1- √ 1-4pqu 2 2pu
. We have for all u > 1 real:

F (u, p n ) ≤ G n (u) ≤ F (u, p ∞ )

Quasi-continuous walk

We define that we are in quasi-continuity condition when the values p n does not vary quickly. For example p n = p(αn) where p() is a continuous function and α is a small number. In quasi-continuity condition function G n () is close to the fixed point of the functionnal equation G n (u) = qnu 1-pnuGn(u) , or in other words and uniformly in n:

lim α→0 G n (u) = F (u, p n ) = 1 -1 -4p n q n u 2 2p n u .

Stable walks

We will assume that the random walk probabilities takes a fixed value p ∞ beyond state

N : ∀n ≥ N p n = p ∞ . Therefore for all n ≥ N G n = 1- √ 1-4p∞q∞u 2 2p∞u
. We will investigate the bi-valued random walk where ∀n < N p n = p 1 .

Let u n = 2 √ p n q n and ū = u N . The main singularity of G N (u) is ū since

∀n ≥ N : G n (u) = G N (u) = G N (ū) 1 -D N 1 -u 2 ū2
for some D N , for instance D N = 1. Using recursion 7, we see that u(p ∞ ) is also the main singularity of all G n (u) for n < N

Properties of G n (u) Let s(u) = 1 -u 2 ū2 . Let R n (u) denotes 2p n uG n (u). Notice that R n (u) is even. We have the recursion R n (u) = 2p n q n u 2 1 -pn 2 p n+1 R n+1 (u) (8)
We split

R n (u) = H n (u) -s(u)Q n (u)
with H n (u) and Q n (u) analytical defined and bounded for |u| < |ū| + ε for some ε > 0. Notice that there is only one possible decomposition of R n (u). When n ≥ N , we have

R n = 1 -s(u): H n (u) = Q n (u) = 1.
From recursion (8) we get

H n (u) = 2p n q n u 2 1 1-pn 2p n+1 Hn+1(u)-pn 2p n+1 √ s(u)Qn+1(u) = 2p n q n u 2 1-pn 2p n+1 Hn+1(u)+ pn 2p n+1 √ s(u)Qn+1(u) (1-pn 2p n+1 Hn+1(u)) 2 +( u 2 ū2 -1)( pn 2p n+1 Qn+1(u)) 2 (9)
Theorem 4. There exists ε > 0 such that H n (u) and Q n (u) are uniformly bounded for |u| < ū + ε.

To prove the theorem we formally identify R N (u) = w and define the bivariate generating function R n (u, w) via the recursion

R n (u, w) = 2p n q n u 2 1 -pn 2qn+1 R n+1 (u, w) (10)
The function R n (u, w) is analytical and has positive Taylor coefficients. We notice the identity

R n (u) = R n (u, 1 -s(u)) .
Lemma 1. There exists ε > 0 such that for all u such that |u| ≤ ū + ε and for all w such that |w| ≤ 1 + ε we have

|R n (u, w)| ≤ 1 + ε.
Proof. We notice that 2p n q n = u 2 n 2 is an increasing function of n. We also

have pn pn+1 ≤ 1. Let n < N , there exist ε > 0 such that if |u| ≤ ū + ε and |R n+1 (u, w)| ≤ 1 + ε, then |R n (u, w)| ≤ 1 + ε.
Indeed from recursion (10):

|R n (u, w)| ≤ |4p n q n u 2 | 1 2 -|R n+1 (u, w)| ≤ |4p n q n u 2 | 1 1 -ε ,
and this we set

|u| < u N -1 thus, |4p n q n u 2 | ≤ |u| 2 u 2 N -1 ≤ 1 -ε 2 , for some ε such that |u| ≤ ū + ε.

INRIA

Proof of Theorem 4 Now we have to express H n (u) and

Q n (u) with R n (u, w). Let define R n (u, 1 -x) we denote H n (u, y) the x-even part of R n (u, 1 -x), namely H n (u, x 2 ) = 1 2 (R n (u, 1 -x) + R n (u, 1 + x)) and Q n (u, y) thex-odd part of R n (u, 1 -x), namely -1 2x (R n (u, 1 -x) -R n (u, 1 + x))
. Both are analytical in u and y and we have the relation

H n (u) = H n (u, 1 -u 2 ū2 ) Q n (u) = Q n (u, 1 -u 2 ū2 )
It remains to prove that H n (u, x 2 ) and Q n (u, x 2 ) are bounded, which is given by the integral representation

H n (u, y) = 1 2iπ zH n (u, 1 -z) dz z 2 -y Q n (u, y) = 1 2iπ H n (u, 1 -z) dz z 2 -y
with appropriate integral loops. This last identities give the formal proof that the H n (u) and Q n (u) are analytical and uniformly bounded for |u| ≤ ū + ε for some ε > 0. This terminates the end of the proof of theorem 4.

Theorem 5. There exists a neighborhood of ū such that the following holds for all n ≥ N : (i) log(G n (u)) exists, (ii) let the decomposition log G n (u) = h n (u) -s(u)q n (u), functions h n (u) and q n (u) are uniformly bounded, (iii) q n (u) decays exponentially when n decreases.

Lemma 2. For all u such that |u| ≤ ū + ε and for all |w| ≤ 1 + ε, and for all n < N we have

|R n (u, w)| ≤ min{4p n ū2 , 1}(1 + ε)
Proof. We just reuse recursion (10) and notice that when n < N

|R n (u, w)| ≤ pnqn p N -1 q N -1 4p N -1 q N -1 |u| 1 2-|Rn+1(u,w)| ≤ pnqn p N -1 q N -1 (1 + ε)
We end the proof of the lemma with the fact that pnqn p N -1 q N -1 ≤ pnqn p∞q∞ = 4p n q n ū2 ≤ 4p n ū2 .

Corollary 1. For all u such that |u| ≤ ū + ε and for all n ≤ N the functions 1 pn H n (u) and 1 pn Q n (u) are uniformly bounded.

Lemma 3. For all u such that |u| ≤ ū:

|R n (u)| ≤ pn p∞ ≤ 1 Proof. When |u| ≤ ū we have |G n (u)| ≤ G n (ū) ≤ 1 2p∞ ū .
Lemma 4. For all ε > 0, there exists a neighborhood of ū such that Q n (u) exponentially decays when n decreases and

|Q n (u)| ≤ Q 1-ε n (ū).
Proof. We rewrite recursion (8) with

H n (u) = 1 2pnqnu 2 H n (u) Hn (u) 1 -pn 2pn+1 H n+1 (u) Q n (u) = 1 4pn+1qnu 2 H n (u) Hn (u)Q n+1 (u) with Hn (u) = H n (u) + s(u)Q n (u). Furthermore, since 1 pn R n (u) = 1 pn H n (u) -s(u) 1 pn Q n (u), 1 pn H n (u) and 1 pn Q n (u)
, are uniformly bounded for all n ≤ N , thanks to Cauchy theorem, they are also uniformly continuous. Since s(ū) = 0, For any ε > 0 there exists a neighborhood of ū such that |R n (u)| < pn p∞ (1 + ε), uniformly for all n ≤ N .

RR n°0001

Similarly, since Rn (u) = R n (u) +2 s(u)Q n (u) we also have a neighborhood of ū where | Rn (u)| ≤ pn p∞ (1 + ε). Consequently for u in this neighborhood (assuming |u -ū| < εū), we have

|Q n (u)| = 1 4pn+1pn|u| 2 R n (u) Rn (u) |Q n+1 (u)| ≤ 1 4pn+1pn|u| 2 p 2 n p 2 ∞ (1 + ε) 2 |Q n+1 (u)| ≤ 1 4pn+1pn ū2 p 2 n p 2 ∞ (1 + ε) 2 ū2 |u| 2 |Q n+1 (u)| ≤ 4p∞q∞ 4pn+1qn p 2 n p 2 ∞ (1 + ε) 4 |Q n+1 (u)| =
|u-ū| ≤ ε such that |Q n (u)-Q n (ū)| ≤ Q n (ū) (exp((N -n)|u -ū|)β) -1). Lemma 5. We have H n (u) = R n (ū) > 2p n ū.
Proof. we have R n (u) = 2p n ūG n (ū) and also the fact that G n (ū) ≥ 1.

End of proof of theorem 5 Since 1 pn R n (ū) is uniformly bounded from below and 1 pn R n (u) is uniformly bounded and continuous around ū, then there exists a neighborhood of ū where 1 pn R n (u) and 1 pn H n (u) are non zero and uniformly bounded from above and from below. If this neighborhood is a simple disk, then log 1 pn R n (u) and log 1 pn H n (u) exist. We also have the identity Notice that q n (u) is of order Qn(u) Hn(u) and decays exponentially as Q n (u) when n decreases. This terminates the proof of theorem 5

log 1 p n R n (u) = h n (u) -s(u)q n (u) with h n (u) = log 1 pn H n (u) + 1 2 log(1 -s(u) Q 2 n (u) H 2 n (u) ) q n (u) =
We denote D n = q n (ū) we have

D n = p n q n G n (ū)G n+1 (ū)D n+1
Quasi-continuous concave walk We have the assymptotic estimate.

log

F n (u) = n j=1 log G j (u) = 1 α αn 0 log F (u, p(x))dx + O(1)
and

log D n = n j=1 log p j q j G j (ū)G j+1 (ū) = 1 α ∞ αn log p(x) q(x) F 2 (ū, p(x)) dx + O(1) .
INRIA Figure 3: actual potential (dashed) and apparent asymptotic potential (plain) for a concave walk with p n = 0.48 exp(-100 n 2 ) N = 100, t = 10, 000, ρ = exp(0.0005)ū -1 .

Case where ρū > 1

This is the simplest of the three cases. The quantity 1 ρ is the main singularity and lead to a simple pole. Quantity ū is the second singularity whose contribution will be detailed in the next section. If we assume that 1 ρ < z < ū we get

r n (t) = F n ( 1 ρ )ρ -t + O(F n (z) z -t 1 -ρz ) With F n (u) = j=n j=1 G j (u) we get pn (t) = p n 1 ρ G n+1 ( 1 ρ ) + O((ρz) -t )
In quasi-continuity condition we have or all u < u n : G n (u) = F (u, p(αn))+O(α) and therefore we get the result we had for uniform random walk that is pn (t) = p(αn)

1 ρ F ( 1 ρ , p(αn)) + O(α) + O((ρz) -t ) .
Figure 3 shows the apparent potential when ρū > 1.

Case where ρū < 1

Theorem 6. When ρū < 1, with n = o( √ t)we have the asymptotic estimate

r n (t) = √ π Fn(ū) (n+t) 3 2
(p∞q∞)

1 4 (ū) t j=n j=1 D j (g ρ (ū) + (-1) n+t g ρ (-ū)) (1 + O( N n+t )) + O((1 + ε) n z -t ) With F n (u) = j=n j=1 G j (u),.
Proof. We define z = ū(1 + ε) which is the upper-limit of the radius where |u| ≤ z implies |R n (u)| < 1 + ε. In this section we assume that 1 ρ > z, i.e. the disk |u| ≤ z does not contain any other singularoties than ±ū for F n (u)g ρ (u).

RR n°0001

P. Jacquet

We have

F n (u) = j=n j=1 G j (u) = 1 u n exp   j=n j=1 h j (u) -s(u)q j (u)  
We have h n (ū) = log(ūG n (ū)). Let D n = q n (ū) = Qn(ū) Hn(ū) . We have the expression

D n = p n q n G n (ū)G n+1 (ū)D n+1 Which gives D n = j=N j=n pj qj G j (ū)G j+1 (ū). Notice that pn qn G n (ū)G n+1 (ū) ≤ 1 since pn qn ≤ p∞ q∞ p n q n G n (ū)G n+1 (ū) ≤ p ∞ q ∞ F 2 (ū, p ∞ ) = p ∞ q ∞ 1 4(p∞) 2 ū2 = 1
Our aim is to find an accurate estimate of r n (t) via the Cauchy formula

r n (t) = 1 2iπ R j (u) p j du u n+t g ρ (u)
As in the previous section we have the estimate

r n (t) = I n (t, z) + J n (t, z) + O((1 + ε) n z -t )
The key of our analysis is the estimate of I n (z, t) and J n (z, t). This is an integration with the factor u n F n (u) = exp( j=n j=1 h j (u) -s(u)q j (u)). By doing the change of variable u = (1 + v n+t )ū we get

1 u t F n (u) = 1 ūn+t exp( j=n j=1 h j (ū) + 1 n+t O 1 (v) - √ -2v Dj √ n+t + √ -2v 1 (n+t) 1 2 O 2 (D j (e (N -j)|u-ū|β -1)))e -v = 1 ūt j=n j=1 G j (ū) 1 - √ -2v P j=n j=1 Dj √ n+t + n n+t O ′ 1 (v) + √ -2v 1 (n+t) 3 2 j=n j=1 O ′ 2 (vD j (N -j)) e -v
Notice that O 1 (v) and O ′ 1 (v) are both analytic in v with the domain of definition including the integration paths.

Therefore expression I n (t, z) we have

I n (t, z) = 1 2iπ z ū exp j=n j=1 h j (u) + i u 2 ū 2 -1q j (u) -exp j=n j=1 h j (u) -i u 2 ū 2 -1q j (u) g ρ (u) du u n+t+1 = ū-t π j=n j=1 G j (ū) j=n j=1 D j √ 2ve -v + j=n j=1 O ′ 2 (D j ) N (n+t) 3 2 + O((1 + ε) n z -t ).
Notice that the term in O ′ 1 (v) n n+t disappears when we substract the term in -i u 2 ū2 -1:

r n (t) = √ π Fn(ū) (n+t) 3 2
(p∞q∞)

1 4 (ū) t j=n j=1 D j (g ρ (ū) + (-1) n+t g ρ (-ū)) (1 + O( N n+t )) + O((1 + ε) n z -t ) INRIA With F n (u) = j=n j=1 G j (u)
, we get the following evaluation for the apparent repulsion:

pn (t) = p n G n+1 (ū)u N j=n+1 j=1 D j j=n j=1 D j (1 + O( N n + t ))
In quasi-continuity condition we can identify G n (u) with F (u, p n ) and

D n+1-k = ( pn qn F 2 (u, p n )) k D n+1 . This leads to j=n+1 j=1 D j = 1 1-pn qn F 2 (u,pn) D n+1 and j=n j=1 D j = pn qn F 2 (u,pn) 1-pn qn F 2 (u,pn) D n+1 lim α→0,t→∞ pn (t) = ū F (ū, p n ) q n .
It turns out that the black hole is simply repulsive since pn (t)

≥ (u N ) 2 2p n q n = 1 2 1 + 1 -( ū un ) 2 ≥ 1 2
, and the closer we are to the black hole, the stronger is the repulsion. When n > N we get naturally pn (t) ≈ 1 2 : the random walk is apparently neutral beyond the state s N where the coefficients are stable.

In the limit case where u N ≈ 1, we would have pn (t) ≈ q n : the random walk is apparently strictly reversed.

Corollary 3. We have r n (t) = P (T n = t)( 1 1-ρū + (-1) n+t 1+ρū )(1+O( N n+t ))+P (T n > t)

Case where ūρ ≈ 1

We consider the case where ρ ≈ ū-1 but with ū > 1.

Case 1 ρ < ū Theorem 7. For n = o( √ t) we have r n (t) = ρ t F n ( 1 ρ ) + ū-t √ πF n (ū) P j=n j=1 Dj (n+t) 3 2 (p ∞ q ∞ ) 1 4 (g ρ (ū) + (-1) n+t g ρ (-ū)) (1 + O( N n+t )) +O((1 + ε) n z -t ) .
Proof. We assume that 1

ρ -ū = 1 o(t) = o( 1 √ t )
. It suffice to add the contribution of both 1 ρ and ū:

r n (t) = ρ t F n ( 1 ρ ) + ū-t √ πF n (ū) P j=n j=1 Dj (n+t) 3 2 (p ∞ q ∞ ) 1 4 (g ρ (ū) + (-1) n+t g ρ (-ū)) (1 + O( N n+t )) +O((1 + ε) n z -t )
Notice that the term in ū-t is smaller than the term in ρ t .

Case 1 ρ > ū Theorem 8. Assuming ρū = 1 -1 o(t) , when n = o( √ t)
we have the estimate:

rn(t) Fn(ū) = ρ t + ū-t √ π P j=n j=1 Dj (n+t) 3 2 (p ∞ q ∞ ) 1 4 (g ρ (ū) + (-1) n+t g ρ (-ū)) +O( 1 q 1 ρ -ū )ρ t j=n j=1 D j
Proof. This case is interesting because the second main singularity 1 ρ stands right in the integration path of I n (t, z), and g ρ (u) becomes singular at u = 1 ρ . To remove this annoying singularity we bend the integration path of I n (t, z) so that it avoids the point 1 ρ and therefore the singularity at u = 1 ρ becomes a simple pole. Anyhow the detour will introduce a correction term of order F n (ū)g ρ (ū)

n j=1 D j 1 -1 (ρū) 2 ρ t . That is an error term in F n (ū)ρ t O( 1 q 1 ρ -ū ) n j=1 D j .
We assume that

= ρ t + ū-t √ π P j=n j=1 Dj (n+t) 3 2 (p ∞ q ∞ ) 1 4 (g ρ (ū) + (-1) n+t g ρ (-ū)) +O( 1 q 1 ρ -ū )ρ t j=n j=1 D j Notice that g ρ (ū) = O( 1 1 ρ -ū ) = o(t)
and the error term is negligible in front of the term in ū-t . Since

F n ( 1 ρ ) = F n (ū)(1 + O( n √ t )) and
pn (t) ≈ p n 1 ρ F ( 1 ρ , p n+1 ) ρ t + ū-t √ π(p ∞ q ∞ ) 1 4 (n + t) -3 2 1 1-pn qn F 2 (ū,pn) D n+1 (g ρ (ū) + (-1) n+t g ρ (-ū)) ρ t + ū-t √ π(p ∞ q ∞ ) 1 4 (n + t) -3 2 pn qn F 2 (ū,pn) 1-pn qn F 2 (ū,pn) D n+1 (g ρ (ū) + (-1) n+t g ρ (-ū))
.

We cannot say that any of the two terms in ρ t and in ū-t is negligible in front of the other one.

Corollary 4. In quasi continuous condition there is a state B such that the black hole is attractive before this state and attractive beyond and

B = z α such that ∞ z log p(x) q(x) F 2 (ū, p(x)) dx = α log ρ t ūt t 3 2 (1 -ρū) .
Proof. We observe that if we define state B such that

ρ t ≈ ū-t √ π(p ∞ q ∞ ) 1 4 1 1 -p B q B F 2 (ū, p B )
D B+1 (g ρ (ū) + (-1) n+t g ρ (-ū)) .

When n < B then the term in ρ t will be preponderant and in this case

pn (t) ≈ p n 1 ρ F ( 1 ρ , p n )
in other word the black hole is attractive.

When n > B, then pn (t) ≈ q n 1 ρF ( 1 ρ , p n ) in other words, the black hole is repulsive beyond state B. Notice that the change of mode is sharp: there is a brisky change of the value of pn (t) in a small set of contiguous states, leading to an edge in the apparent random walk potential (see figure 4 andfollowing).

Notice that pn (t) → 1 2 when n increases: the random walk is asymptotically neutral.

Ignoring O(α) terms, the change of mode occurs on state B = z α such that ∞ z log p(x) q(x) F2 (ū, p(x)) dx = α log ρ t ūt t Figures 4 and5 shows the apparent potential of a concave stable walk with different ρ < ū-1 .

INRIA Figure 4: actual potential (dashed) and apparent asymptotic potential (plain) for a concave walk with p n = 0.48 exp(-100 n 2 ) N = 100, t = 10, 000, ρ = exp(-0.002)ū -1 (left), ρ = exp(-0.001)ū -1 (right).

Figure 5: actual potential (dashed) and apparent asymptotic potential (plain) for a concave walk with p n = 0.48 exp(-100 n 2 ) N = 100, t = 10, 000, ρ = exp(-0.004)ū -1 (left), ρ = exp(-0.006)ū -1 (right).

Generalized concave walks

In this subsection we don't consider anymore that the random walk coefficient are stable beyond a fixed step N . In this case we simply assume that lim p n = p ∞ and we denote ū = u(p ∞ ).

The main difficulty is in the convergence of the decomposition functions R n (u) = H n (u) -s(u)Q n (u). Therefore we assume that the series √ p ∞ -p n converge. We can prove that there exists ε > 0 and N such for all u with |u| < ū(1 + ε) and for all n > N :

R n (u, v) = v + O( j≥n p ∞ -p j )
A more thorough analysis shows that lemma 1 can be extended to the statement that when |u| < u N the following bounding condition holds:

|R N (u, v)| ≤ 1 + 1 -|u| 2 u 2 N . To this end we assume that 1 -|u| 2 u 2 N ≤ (1 -ε) 1 - ū2 u 2 N ≤ (1 -ε) 2(p ∞ -p N )). But j≥n p ∞ -p j = o( √ p ∞ -p n . Therefore by letting N large enough such that |R N (u, v)| ≤ 1 + 1 -|u| 2 u 2 N
, we can export the results of stable random concave walks to general concave walk.

Non unitary Gravitational walks

We call gravitational walk a concave walk where p ∞ = 1 2 . Or, in other words ū = 1. For example p n = 1 2 -α 2 n 2 . In this case we have 4p n q n which converge to one as (p ∞ -p n ) 2 .

Theorem 9. We have the estimate r n

(t) = ρ t + P (T n > t) + P (T n = t)( 1 1-ρ + (-1) n+t 1+ρ )(1 + O( N √ n+t )) with P (T n = t) = π/2 j=n j=1 D j t -3 2 1 + O( N n+t ) P (T n > t) = √ 2π j=n j=1 D j t -1 2 1 + O( N n+t ) .
Proof. The analysis of the stable gravitational walk and the decomposition

F n (u) = j=n j=1 G j (u) and R n (u) = 2p n uG n (u) = H n (u) -s(u)Q n (u)
remains with the asymptotic estimates. Nevertheless the contribution of g ρ (ū) becomes singular when ū = 1. In this case we have to develop the asymptotics of P (T n > t) separately and this become the preponderant term. It comes that r

n (t) = ρ t + P (T n > t) + P (T n = t)( 1 1-ρ + (-1) n+t 1+ρ )(1 + O( N √ n+t )) with P (T n = t) = π/2 j=n j=1 D j t -3 2 1 + O( N n+t ) P (T n > t) = √ 2π j=n j=1 D j t -1 2 1 + O( N n+t )
And we have

D n = p n q n D n+1 = j=∞ j=n p j q j .
Notice that:

D n = exp (2(V n -V ∞ )) .
The transition to generalized gravitational walks is somewhat different because 4p n q n converges to one like (p ∞ -p n ) 2 since p ∞ = 1 2 instead of converging

INRIA

Figure 6: actual potential (dashed) and apparent asymptotic potential (plain) for a gravitational walk with p n = 1 2 exp(-1000 n 2 ), t = 10 8 , ρ = exp(-50.10 -8 ) (left), ρ = exp(-200.10 -8 ) (right), in green we display the average value between actual and apparent potential. to 4p ∞ q ∞ like p ∞ -p n when p ∞ < 1 2 . Therefore the condition of the transition to the asymptotics of the previous section is now that the series in p ∞ -p n converges (instead of the series in √ p ∞ -p n ).

We consider the case where ρ is very close to one

Corollary 5. When 1 o(t) < 1 -ρ < o( 1 √ t ) we have r n (t) = ρ t + P (T n > t)(1 + o(1))
Proof. For this model we get 1 1-ρ P (T n = t) = o(P (T n > t)) and since F n (1) = 1:

r n (t) = ρ t + P (T n > t)(1 + o(1))
In other words the non unitary effect is equivalent to end pay back model: if the rabbit reaches the black hole before time t, the unitary effect is ρ t , otherwise it is 1 where we should have r n (t) = ρ t + P (T n > t)(1 -ρ t ).

The quasi continuous condition brings a similar conclusion as in the previous section. Defining The black hole is attractive until state B = z α with pn (t) ≈ p n : the attraction is unchanged. Beyond state B the random walk is repulsive with pn ≈ q n :. the attraction is completely reversed beyond that state. See figure 6 In quasi continuous situation, we denote V (y) = y 0 1 2 log( q(x) p(x) )dx. For x = αn it turns out that V n = 1 α V (x) + O(1) and D n = exp(-2 α (V (x) -V (∞)). Similarly . Denoting ρ = exp( β t ), we have

r n = e -β + √ 2π √ t exp( 2 α (V (x) -V (∞)) 1 -p(x)
q(x)

1 + 1 2β + O( 1 t
) .

Figure 7: actual potential V (x) (dashed) and apparent asymptotic potential Ṽ (x) (plain) for a gravitational walk with p n = 1 -1 1+exp(-1 1+x 2 ) , t = 10 90 , β = 10 20 , α = 1 2 .10 -20 , in green we display the average value between actual and apparent potential.

Assuming t and 1 α large the change of mode occurs for z such that

β = 1 2 log t + 2 α (V (∞) -V (z)) + log(1 - p(z) q(z) )
We can investigate a special case where the computations are tractable. For example when log( q(x) p(x) ) = 1 1+x 2 , namely p n = 1 -1 1+exp(-1 1+x 2 ) . In this case we have V (x) = 1 2 arctan(x). Denoting Ṽ (x) = α , figure 7 displays the actual and apparent potential for this case with parameters tuned to galactic orders of magnitudes: t = 10 90 , α = 1 2 .10 -20 , β = 10 20 . We get z ≈ 1.7. The bimodal aspect of the apparent gravitationnal, and in particular the exact reversion of the random walk beyond critical distance z give some reminiscence about the mysterious dark energy effect. Since 1998, it has been noted that the expansion of the universe is in acceleration, fact that contradicts the usual decelerated expansion scheme predicted by general relativity. In order to explain this discrepancy, astrophysicist have imagined a so-called dark energy that contributes to the acceleration of the expansion. In classic newtonnian physics, the dark energy is equivalent to a medium that excerts a repulsion that is twice is gravitational attraction, a kind of force which does not exist in the standard models at least at low energy. In the gravitational walk model the dark energy effect would be exactly the effect of random walk reversion of low energy usual matter beyond the critical distance, assuming that gallactic black holes are non unitary.

It should be noted that in the model presented in figure 7 we would have a non unitary effect of less than 10 -70 per time unit, inside a black hole. It means that the impact of the non unitary effect within the critical distance is not measurable, and that only the extremely large black hole lifetimes would make it apparent beyond the critical distance. It should also be noted that the repulsion is not due to a force in the physical sense, but would be due to a weird effect in future trajectory weight summation in the random walk.

INRIA

Figure 2 :

 2 Figure2: actual potential (dashed) and apparent asymptotic potential (plain) for p = 0.2 for ρ = 0.79 (left) and ρ = 0.3 both smaller than 2 √ pq = 0.8.

  ε) 4 |Q n+1 (u)| ≤ q∞ qn pn p∞ (1 + ε) 4 |Q n+1 (u)| Therefore Q n (u) decays exponentially like j=N -1 j=n q∞ qj pj p∞ (1 + ε) 4 as soon as q∞ qn pn p∞ (1 + ε) 4 < 1. Corollary 2. There exists ε > 0 and a real number β such that for all n ≤ N when

  qn F 2 (ū,pn) D n+1 in quasi continuous condition, we get (removing the error term)

= α log ρ t t 1 2

 2

RR n°0001

(1 -ρū)


Stephanie Reynaud

Antoine Aynard

Bruno Grassl
email: bruno.grassl@univ-pau.fr

Julien Gigault
email: julien.gigault@takuvik.ulaval.ca>

NANOPLASTICS : FROM MODEL MATERIALS TO COLLOIDAL FATE

Keywords: Nanoplastics, model materials, colloidal properties, stability, aggregation

Nanoplastics are pieces of plastic debris of mixed shape and chemical composition, which can be present in the air, soil, freshwater bodies, seas, biota, and our diet. We know little about the prevalence of nanoplastics and the risks they pose to the health of living organisms; recent studies of plastics in the environment have not significantly contributed information about this crucial topic. In this article, nanoplastics are presented as colloids.

Analysing their colloidal properties shows that their fate is governed by interfacial

Introduction

The spread and potential risks of plastic debris is an international, global concern.

Debris degrades in the environment into ever-smaller fragments, which increases the surface-to-volume ratio. The smallest pieces, microplastics and nanoplastics, are thought to affect ecosystem health and biodiversity conservation via multiple pathways [START_REF] Gigault | Current opinion: What is a nanoplastic?[END_REF][START_REF] Hartmann | Are We Speaking the Same Language? Recommendations for a Definition and Categorization Framework for Plastic Debris[END_REF]. It is known that debris size and risk to human health are inversely proportional, but there is not yet sufficient scientific knowledge of nanoplastics to establish relevant policies for risk assessment.

Due to the intrinsic properties of nanoplastics, especially in the aqueous phase where they behave as a colloid, research strategies (analytical methods, risk assessment, and sampling) need to be developed differently than for macroplastics and microplastics. The challenges increase as particle size decreases: nanoplastics interact more, or differently, with biological systems, and multiple research gaps must be adequately addressed to build a sustainable future.

Weathered macroplastics and microplastics from mismanaged waste are undoubtedly the most common source of nanoplastics, but material wear, such as of synthetic fabrics or car tire abrasion, can also lead to environmental nanoplastics [START_REF] Boucher | Primary microplastics in the oceans: A global evaluation of sources[END_REF]. The literature does not yet report any effective methods to trace the origin, source, or transport of microplastics, let alone nanoplastics. Finding such methods is challenging due to the diversity of the chemical makeup of the polymer matrices and the variety of additives that are used. Risk assessment for environmental microplastics and nanoplastics is challenging due to the lack of reference materials and validated detection methods, particularly in complex biological matrices.

Nanoplastics have been detected in seawater, but not quantified. It is known, however, that the smaller the plastics, the larger the quantities in terms of number of entities [START_REF] Phuong | Microplastic abundance and characteristics in French Atlantic coastal sediments using a new extraction method[END_REF]. To invent new nanoplastic sampling and purification techniques, the scientific community is now using model materials of nanoplastic particles to establish, optimize, and apply novel analytical methods and sample preparation protocols to characterize and quantify nanoplastics in various matrices. This review aims to present nanoplastics onto the context of plastic pollution, scientific issues associated to this specific kind of plastic debris from the model material to their behavior as a colloid. It starts by briefly recalling the plastic pollution context with the issues related to microplastics, and why we need a distinct research category for nanoplastics. We then consider problems in defining experimental conditions for nanoplastics. Nanoplastic model materials, with their interfacial and colloidal properties, are intrinsic to this research. The review finishes with the phenomena of homo-and heteroaggregation due to the colloidal and interfacial aspects of nanoplastics.

From microplastics to nanoplastics: context and issues.

Microplastics and nanoplastics have radically different chemical behaviours. The category of nanoplastics was defined recently in response to studies conducted on microplastics, which revealed that microplastics smaller than 1 µm behaved so differently from larger microplastics (1 µm to 5 mm) that they needed to be classified as a completely separate category [START_REF] Gigault | Current opinion: What is a nanoplastic?[END_REF][START_REF] Hartmann | Are We Speaking the Same Language? Recommendations for a Definition and Categorization Framework for Plastic Debris[END_REF][START_REF] Gigault | Nanoplastics are neither microplastics nor engineered nanoparticles[END_REF]. Studies and reviews on microplastics have covered various environmental aspects [START_REF] Da Rocha | Microplastic pollution in Southern Atlantic marine waters: Review of current trends, sources, and perspectives[END_REF][START_REF] Harris | The fate of microplastic in marine sedimentary environments: A review and synthesis[END_REF][START_REF] Rillig | Microplastic in terrestrial ecosystems[END_REF][START_REF] Zhang | Atmospheric microplastics: A review on current status and perspectives[END_REF] .

The scientific community classifies microplastics according to their origins, either as primary or secondary microplastics. Primary microplastics (PMPs) are plastic items that are deliberately manufactured at microscopic sizes (below 5 mm). These PMPs are used in exfoliating cosmetics, toothpaste, makeup, synthetic clothing, compressed air sandblasting processes, or as vectors of drugs in medicine [START_REF] Auta | Distribution and importance of microplastics in the marine environment: A review of the sources, fate, effects, and potential solutions[END_REF]. This category also includes plastic granules of 2-5 mm diameter; often used as industrial raw material to generate plastic products [START_REF] Boucher | Primary microplastics in the oceans: A global evaluation of sources[END_REF]. Personal care products such as plastic microbeads are intentionally poured into wastewater, and are now banned in some countries to reduce this source of plastic pollution. Other PMPs found in the environment come from unintentional losses along the whole plastic value chain (weathering or leaching at open-air waste reception centers, accidental spills, losses during the transport of materials (plastic pellets) or through abrasion (such as tire wear on roads). PMPs can be deposited in rivers through sewage water (microfibre clothing fibres in washing machines) since wastewater treatment plants cannot separate them from wastewater. This is a way to reach another environmental compartment since microplastics may also be concentrated in the sludge during the water treatment and used in many countries as agricultural fertilizer [START_REF] Alimi | Microplastics and Nanoplastics in Aquatic Environments: Aggregation, Deposition, and Enhanced Contaminant Transport[END_REF][START_REF] Cole | Microplastics as contaminants in the marine environment: A review[END_REF].

Secondary micro-plastics (SMPs) arise from mismanaged or dumped domestic and commercial plastic wastes, which are altered by physical, chemical, and biological processes in the environment, whether on soils or in surface water [START_REF] Cole | Microplastics as contaminants in the marine environment: A review[END_REF]. Exposure to sunlight, and in particular ultraviolet light, causes plastic photodegradation and surface oxidation, which may be followed by a thermo-oxidative process [START_REF] Andrady | Microplastics in the marine environment[END_REF]. The degradation first occurs at the outer surface as revealed by a change of color (yellowing) and the appearance of microcracks as the surface becomes brittle [START_REF] Ter Halle | Understanding the Fragmentation Pattern of Marine Plastic Debris[END_REF]. SMPs will then be liable to fragment under the effects of abrasion, waves, and turbulence.

While the sample collection reported that macroplastics are almost exclusively composed of polyolefins with polyethylene (PE) / polypropylene (PP) with the ratio 80%/20%, microplastics collection exhibit a larger diversity. A microplastic collection in the north Atlantic subtropical gyre was composed of 73% PE, 13% PP, 8% PVC (polyvinylchloride), 2% PS (polystyrene), and 1% PET (polyethylene terephthalate). This difference in term of composition versus the size of debris, may be attributed to several factors as the sample collection methods, in terms of items numbers where microplastics represent 80-90% of the collected items, the rest stands for meso-and macroplastics and in terms of the location (geographical and within the water column) where the collection has been done [START_REF] Ter Halle | Understanding the Fragmentation Pattern of Marine Plastic Debris[END_REF].

In the well-studied marine environment, the results of sampling campaigns in ocean gyres and models estimating the amount of microplastics differ, showing that only 1% of the expected quantity is observable [START_REF] Cózar | Plastic debris in the open ocean[END_REF]. Several hypotheses have been suggested to explain this significant difference. The sampling method comes first; indeed, the regulatory mesh size of the plankton net is 330 μm, lower mesh size down to 64 μm has been reported for specific expeditions missing the smaller debris [START_REF] Figueiredo | Suspended microplastics in a highly polluted bay: Abundance, size, and availability for mesozooplankton[END_REF][START_REF] Mattsson | Nano-plastics in the aquatic environment[END_REF]. Consequently, the lower size limit for sampled microplastics varies between researchers due to the protocol [START_REF] Frias | Microplastics: Finding a consensus on the definition[END_REF]. The sampling procedure for microplastics in the water column misses the sinking fraction. As they were not properly sampled or even not sampled at all, sub-micron plastic particles were scarcely studied. Their quantification is still not possible as there are not yet any adequate detection tools.

Weathering of macroplastics and microplastics yields debris of sub-micron sizes [START_REF] Gigault | Marine plastic litter: the unanalyzed nano-fraction[END_REF][START_REF] Lambert | Characterisation of nanoplastics during the degradation of polystyrene[END_REF][START_REF] Lambert | Formation of microscopic particles during the degradation of different polymers[END_REF] that are expected to be even more prevalent than their bulk counterparts. The mass and volume concentration of this small fraction dispersed in water is extremely low but incredibly high in terms of surface area as the surface area to the volume ratio gets smaller as the cell gets larger being proportional to 1/D, with D the diameter. Up to now, this small category represents a low mass concentration that remains below the quantification limit of analytical methods, and the lack of withdrawal pre-concentration established procedures do not allow studies using "real, or even representative" items from surface water. Nevertheless, their presence was detected in the North Atlantic Gyre to prove their ability to accumulate in the oceans, just like macro and micro-plastics [START_REF] Ter Halle | Nanoplastic in the North Atlantic Subtropical Gyre[END_REF]. They also were detected in the sand [START_REF] Davranche | Nanoplastics on the coast exposed to the North Atlantic Gyre: Evidence and traceability[END_REF] and soil [START_REF] Wahl | Nanoplastic occurrence in a soil amended with plastic debris[END_REF] more recently. At last but not least, little work has been done so far onto the removal of nanoplastics. Due to the size of these debris, physical withdrawal is not feasible, however recent study reported interesting strategy using photocatalytic mineralization in aqueous media [START_REF] Allé | Efficient photocatalytic mineralization of polymethylmethacrylate and polystyrene nanoplastics by TiO2/β-SiC alveolar foams[END_REF].

Nanoplastics: a distinct category

The size of plastic debris impacts their behaviour in the environment and their potential toxicity to living organisms. Nanoplastics are identified as emerging contaminants of concern [START_REF] Koelmans | Proxies for nanoplastic[END_REF]. Unlike larger particles, nanoplastics can exhibit colloidal behavior [START_REF] Gigault | Current opinion: What is a nanoplastic?[END_REF].

Nanoplastics are generated by the degradation and fragmentation of larger plastic items in the environment, and should not be confused with engineered nanoparticles <100 nm [START_REF] Gigault | Nanoplastics are neither microplastics nor engineered nanoparticles[END_REF].

A recent review highlighted the role of the chemical composition, shape, and interface on the distinct fate of nanoplastics [START_REF] Kihara | Reviewing nanoplastic toxicology: It's an interface problem[END_REF].

Scientists, policymakers, and the public are becoming increasingly concerned about the prevalence of nanoplastics and the uncertainties surrounding their impacts, hazards, and risks to our environment and human health. Some of the evidence remains uncertain, and it is linked directly to their intrinsic nature: differences in polymer nature, item size (continuum size up to 1 µm), shape (spheroids, cylindrical pellets, fragments, flakes…), chemical additives (release or lixiviation), concentrations, measurements, fate.

Nanoplastics should also be considered as a chemical cocktail coming from their production and environmental ageing. They may act as a pollutant carrier, as they have low-polarity surfaces that interact favourably with pollutant chemicals [START_REF] Mitrano | Placing nanoplastics in the context of global plastic pollution[END_REF][START_REF] Hartmann | Microplastics as vectors for environmental contaminants: Exploring sorption, desorption, and transfer to biota[END_REF]. This may even be more pronounced with nanoplastics than microplastics due to a larger surface/volume ratio. The nanoplastic-pollutant cocktail may interact with and be ingested by organisms, and may either worsen or mitigate the pollutant exposure. The physical one should not be excluded from this potential chemical risk, such as inflammation, stress, and the gastrointestinal compartment [START_REF] Kokalj | Quality of nanoplastics and microplastics ecotoxicity studies: Refining quality criteria for nanomaterial studies[END_REF].

The colloidal properties of nanoplastics are another critical factor in why they may have higher toxicities than microplastics [START_REF] Bouwmeester | Potential Health Impact of Environmentally Released Micro-and Nanoplastics in the Human Food Production Chain: Experiences from Nanotoxicology[END_REF]. As they are not affected by the density of their constituent plastic, they stay in the water column more easily, and may statistically have more interaction with pollutants and be carried by sea currents over long distances before being eventually ingested by marine organisms [START_REF] Maity | Perspectives and challenges of micro/nanoplastics-induced toxicity with special reference to phytotoxicity[END_REF]. Nanoplastic behaviour and toxicity do not depend exclusively on size, and other properties should be evaluated for any risk assessment.

Trojan horse properties of nanoplastics

Virgin polymers, which are inert materials, are made suitable for commodity and specialty applications by the intentional addition of various additives to meet specific technical needs. The generic term "plastic" is defined by the International Union of Pure and Applied Chemistry as "a polymeric material that may contain other substances to improve performance and/or reduce costs" and should not be confused with the term "polymer" [START_REF] Vert | Terminology for biorelated polymers and applications (IUPAC Recommendations 2012)[END_REF]. Plastic additives are divided into four groups [START_REF] Hahladakis | An overview of chemical additives present in plastics: Migration, release, fate and environmental impact during their use, disposal and recycling[END_REF]: 1) functional additives for additional properties (e.g., stabilizers, flame retardants, lubricants, foaming agents, biocides, slip agents, curing agents and plasticizers), 2) colorants (e.g. pigments, soluble azocolorants), 3) fillers: mostly inert, they increase the final product stiffness or hardness and reduce the material cost (e.g. mica, talc, kaolin, clay, calcium carbonate), and 4) reinforcements (e.g. carbon and glass fibres).

From a total of 1550 potential plastic additives, the European Chemical Agency (ECHA) identified more than 400 additives used for plastics in high volumes (i.e., above 100 tons/year) [START_REF]Plastic additives initiative -ECHA n[END_REF]. ECHA developed a model to calculate the release potential of the additives and evaluated 155 of the 400 additives. This inventory is a starting point to decide which substances should be assessed as a matter of priority. Additives that are only incorporated by mixing into the matrix or chemically bonded to the polymer matrix should be considered separately [START_REF] Wagner | Legacy additives in a circular economy of plastics: Current dilemma, policy analysis, and emerging countermeasures[END_REF] . The legislation is increasingly restrictive, and plastic additives, now as known as hazardous substances, are at least screened or regulated, and at best banned [37].

Plastics have been discarded in the environment for decades, including those with persistent organic pollutant additives, and these plastics are now aging and degrading.

Chemical residues (as monomers) or impurities are not within the scope of the definition of additive even though they should have the same behaviour and should be considered an additional potential risk. As mentioned above, nanoplastics are the plastic debris category that exhibits the highest surface-to-volume ratio. The release properties are linked to the diffusion of the additives into the plastic matrix and the partition of the additives from the matrix surface to the medium. As a result, additives embedded in debris are more easily released from nanoplastics than from larger plastic particles.

Nanoplastics' high surface-to-volume ratios mean that they have statistically more contact with other chemical pollutants. Hydrophobic by nature, nanoplastics may behave as efficient sorbents, either by adsorption or absorption, for hydrophobic contaminants [START_REF] Hartmann | Microplastics as vectors for environmental contaminants: Exploring sorption, desorption, and transfer to biota[END_REF]. This is an interfacial phenomenon called the "Trojan horse" property and may result in the transportation of eco-toxic molecules to release them in organisms. Concern about nanoplastics stems more from their potential pollutant carrier behaviour, rather than their intrinsic nature [START_REF] Mitrano | Placing nanoplastics in the context of global plastic pollution[END_REF][START_REF] Hartmann | Microplastics as vectors for environmental contaminants: Exploring sorption, desorption, and transfer to biota[END_REF].

Polychlorinated biphenyls (PCBs) [START_REF] Velzeboer | Strong sorption of PCBs to nanoplastics, microplastics, carbon nanotubes, and fullerenes[END_REF] and polycyclic aromatic hydrocarbons (PAHs) [START_REF] Lee | Sorption capacity of plastic debris for hydrophobic organic chemicals[END_REF] are the most cited persistent organic pollutants that bind to plastic debris. When adsorbed onto nanoplastics, they can be transported along the water column in sediments, or assimilated by organisms, even if they are complexed with organic matter [START_REF] Alimi | Microplastics and Nanoplastics in Aquatic Environments: Aggregation, Deposition, and Enhanced Contaminant Transport[END_REF]. Surface coverage of nanoplastics by contaminants or biological species is called "corona formation" [START_REF] Kihara | Reviewing nanoplastic toxicology: It's an interface problem[END_REF]. Beyond organic pollutants, plastics have a particular affinity with trace metals such as cadmium, lead, bromine [START_REF] Davranche | Are nanoplastics able to bind significant amount of metals? The lead example[END_REF][START_REF] Holmes | Adsorption of trace metals to plastic resin pellets in the marine environment[END_REF][START_REF] Massos | Cadmium, lead and bromine in beached microplastics[END_REF] iron oxides [START_REF] Davranche | Are nanoplastics able to bind significant amount of metals? The lead example[END_REF], copper, zinc, cobalt, nickel, and arsenic [START_REF] Kihara | Reviewing nanoplastic toxicology: It's an interface problem[END_REF][START_REF] Baudrimont | Ecotoxicity of polyethylene nanoplastics from the North Atlantic oceanic gyre on freshwater and marine organisms (microalgae and filter-feeding bivalves)[END_REF]. Depending on the chemical properties of these substances, they may become bioavailable to organisms in the environment. Nanoplastics are dangerous due to their size, as they can penetrate natural barriers [START_REF] Galloway | Ecotoxicological assessment of nanoparticle-containing acrylic copolymer dispersions in fairy shrimp and zebrafish embryos[END_REF][START_REF] Triebskorn | Relevance of nano-and microplastics for freshwater ecosystems: A critical review[END_REF] to enter organisms' bodies.

Nanoplastics and experimental conditions

It is not yet possible to characterize and sample a sufficient quantity of nanoplastics in the environment. Most studies focus on the behaviour and toxicity of plastic nanospheres (PSLs) synthesized at the nanoscale, often commercially available. These PSLs have the advantage of being calibrated in size, known surface area, and, potentially, their known surface functionality. However, they may not be relevant to real environmental nanoplastics, as they are monodisperse, spherical, and do not have a comparable chemical formulation to manufactured plastics for industrial or domestic applications.

Even if we consider that the (eco)toxicological considerations of microplastics and nanoplastics are not in the scope of this paper, (eco)toxicological work and physicochemical studies still do need a relevant and standardized protocol, including the concentration range of nanoplastic-associated pollutants often present in a far different concentration than environmental conditions.

In the case of emerging contamination, it is always difficult to estimate an environmental concentration for performing toxicological tests. Nevertheless, nanoplastic environmental concentrations can be estimated by extrapolating from the known microplastic concentrations, a mass-conserving fragmentation process, and a theoretical 3D fragmentation process. The particulate concentration (n) is then assumed to be proportional to with the inverse of the particle diameter (d) cubed: n∝d -3 [START_REF] Lenz | Microplastic exposure studies should be environmentally realistic[END_REF]. Following this hypothesis, Lenz et al. determined an exponent slightly lower than theoretically expected (n = 3888.d - 2.67 ), which may be caused by size-dependent removal processes, lower dimensional breakdown (i.e., of flakes, sheets, and fibres), or considerable influence of new microplastic input of larger sizes. The lower accuracy in the detection of the smallest items in a sample must be considered. Based on the extrapolation proposed by Lenz, nanoplastics with a hydrodynamic size lower than 1µm have a concentration of 0.14 to 1.4 ng/L (Figure 1) in our oceans, which is about 0.1 to 1% of the 200 Mt of plastic waste already accumulated in the environment [START_REF] Jambeck | Plastic waste inputs from land into the ocean[END_REF]. These concentration levels may appear low in terms of hazards (as a comparison, 4 ng/L represents half a grain of rice in an Olympic swimming pool). Nevertheless, due to their high nanoplastic surface-to-volume ratio, at this concentration and spread over the earth's surface water, the total surface of nanoplastics is similar to those of the earth. Due to the Trojan horse properties, nanoplastics can transport various pollutants from the environment into living organisms, contributing to toxin bioaccumulation [START_REF] Stapleton | Toxicological considerations of nano-sized plastics[END_REF]. This behaviour is mainly governed by the nanoplastic interfacial properties regardless of their aggregation status. Environmental and health issues may be strongly linked to a better understanding of interfacial and colloidal properties of nanoplastics, which are related to the characteristics (shape, size, functionality…) of the nanoplastics themselves [START_REF] Kihara | Reviewing nanoplastic toxicology: It's an interface problem[END_REF].

Research groups have used various nanoplastics as model materials. It is difficult to compare studies in a meaningful way without an official nanoplastic description, so work is underway to establish criteria to properly define nanoplastics and microplastics, especially for those used in toxicity and ecotoxicity studies [START_REF] Kokalj | Quality of nanoplastics and microplastics ecotoxicity studies: Refining quality criteria for nanomaterial studies[END_REF].

Best practices to describe nanoparticles in the future will likely include: 1) reporting the origin of nanoplastics; 2) describing them with shape, size, and size dispersion (followed throughout the test), specific surface area, chemical nature (bulk, additives, impurities, and surface functionality); 3) reporting sample preparation methods (medium physicochemical properties, dispersion mode, nanoplastic concentration, nanoplastic stability within it); 4) identifying the presence/absence of ad/absorbed chemicals (either pollutant or tracer).

At present, producers rarely report the information listed above to describe the PSLs they make. Pikuda et al. were the first to demonstrate that the toxicity of commercial carboxylated PSLs (those commonly used in the research community) is due to additives such as sodium azide (bactericide), and not to the nanoplastics themselves [START_REF] Pikuda | Toxicity Assessments of Micro-and Nanoplastics Can Be Confounded by Preservatives in Commercial Formulations[END_REF].

Nanoplastic model materials

Beyond commercial PSLs, "homemade" nanoplastic model materials have been developed by either the "bottom-up" or "top-down" approaches.

Nanoplastic model materials were obtained by the bottom-up approach by traditional colloidal chemistry methods such as conventional dispersed media (emulsions and nanoprecipitation). Designing such model materials directly in labs allows researchers to adapt the sample to the target study (analytical protocol development, (eco)toxicology with perfect knowledge of the concentration, size, chemical functionality, and content of additives and residuals).

Surfactants are needed for conventional colloidal chemistry (either emulsions or nanoprecipitation). As explained above, those may have detrimental effects on the behaviour studies by being themselves toxic to organisms, as has been shown for sodium dodecyl sulfate, Tween®, and Triton-X®. Organic solvents are also usually needed for nanoprecipitation and should be carefully removed or quantified [START_REF] Balakrishnan | Towards more realistic reference microplastics and nanoplastics: preparation of polyethylene micro/nanoparticles with a biosurfactant[END_REF]. It should be noted using a green surfactant does not solve the toxicity issue; it is due to surfactants' intrinsic ability to adsorb at interfaces, bind to proteins, and solubilize components of membranes.

To overcome this issue, some groups produce their own nanoplastic model materials by performing a so-called "soap-free emulsion" to form nanoplastics without surfactants or additives [START_REF] Kihara | Soft and Hard Interactions between Polystyrene Nanoplastics and Human Serum Albumin Protein Corona[END_REF][START_REF] Wan | A spectroscopic and theoretical investigation of interaction mechanisms of tetracycline and polystyrene nanospheres under different conditions[END_REF]. Following this strategy, Pessoni et al. were able to elaborate nanoplastic model materials free of emulsifiers or additives, including metals, and the model materials are negatively charged with modular properties [START_REF] Pessoni | Soap-and metal-free polystyrene latex particles as a nanoplastic model[END_REF].

Furthermore, tagged nanoplastic model materials may be wanted as toxicology studies to evaluate their biological uptake or follow them in a specific environment [START_REF] Koelmans | Proxies for nanoplastic[END_REF]. Homemade isotopic [START_REF] Al-Sid-Cheikh | Uptake, Whole-Body Distribution, and Depuration of Nanoplastics by the Scallop Pecten maximus at Environmentally Realistic Concentrations[END_REF] or metallic tracer embedded nanoplastics [START_REF] Mitrano | Synthesis of metal-doped nanoplastics and their utility to investigate fate and behaviour in complex environmental systems[END_REF] have been described, fluorescently tagged model materials remain largely reported as Rhodamine or Nile red [START_REF] Jiang | Quantifying the bioaccumulation of nanoplastics and PAHs in the clamworm Perinereis aibuhitensis[END_REF]. Nonetheless, it has been shown that fluorophores which may be toxic to organisms, could desorb from nanoplastics and be accumulated in the tissues of zebrafish larvae [START_REF] Catarino | Use of fluorescent-labelled nanoplastics (NPs) to demonstrate NP absorption is inconclusive without adequate controls[END_REF].

At the same time, the nanoplastics themselves may be removed naturally by the body.

Another approach to yield nanoplastic model materials mimicking the environmental ones leads to the reducing the scale of micro-and macro-plastic samples down to the nanoscale using top-down methods such as laser ablation [START_REF] Magrì | Laser Ablation as a Versatile Tool To Mimic Polyethylene Terephthalate Nanoplastic Pollutants: Characterization and Toxicology Assessment[END_REF], photodegradation [START_REF] González-Pleiter | Secondary nanoplastics released from a biodegradable microplastic severely impact freshwater environments[END_REF], ultrasonication [START_REF] Baudrimont | Ecotoxicity of polyethylene nanoplastics from the North Atlantic oceanic gyre on freshwater and marine organisms (microalgae and filter-feeding bivalves)[END_REF], or mechanical degradation [START_REF] Hadri | Nanoplastic from mechanically degraded primary and secondary microplastics for environmental assessments[END_REF]. Polydisperse nanometric plastic model materials with an irregular shape, surface defects, and an oxidized surface have been obtained in these studies. However, getting large quantities of nanoplastics by these methods can be very difficult and time-consuming, and very few toxicity studies were made possible with these model materials. This research will advance more quickly once we have better knowledge of the nanoplastic concentration in the medium, especially for the (eco)toxicity studies. When reported, really high concentrations are often described.

Colloidal stability of nanoplastics: homo and hetero-aggregation

The stability of nanoparticles, mainly governed by the size and shape of the nanoplastics, is a determining factor in their bioavailability and toxicity [START_REF] Gigault | Nanoplastics are neither microplastics nor engineered nanoparticles[END_REF]. The majority of ecotoxicological studies do not verify the stability of PSLs in the experimental exposure medium. In freshwater studies this does not have a significant impact on the results, because nanoplastics are often stable in freshwater environments. In seawater studies, however, plastic particle size is an essential factor because significant ionic forces cause nanoplastic aggregation [START_REF] Alimi | Microplastics and Nanoplastics in Aquatic Environments: Aggregation, Deposition, and Enhanced Contaminant Transport[END_REF][START_REF] Wang | A review of microplastics aggregation in aquatic environment: Influence factors, analytical methods, and environmental implications[END_REF].

The stability of plastic nanoparticles in the experimental exposure medium has rarely been studied so far. Several studies have measured the hydrodynamic diameter of nanoplastic aggregates in the experimental conditions without specifying measurement time. However, aggregation kinetics allows researchers to define the size characteristics of the particle that will ultimately interact with organisms. When the particles aggregate, the size of the agglomerates increases, and their number decreases accordingly. Changes in these parameters have a considerable impact on the experimental conditions of organisms and, therefore, on the interpretations that may result from them.

As explained in the review of Hotze et al. for classical colloids and nanoparticles [START_REF] Hotze | Nanoparticle Aggregation: Challenges to Understanding Transport and Reactivity in the Environment[END_REF],

nanoplastics, due to their high surface reactivity, interact with natural and anthropogenic colloids such as metallic oxides and other organic molecules and inorganic compounds (clay, silica, etc.) as soon as they diffuse in the environment. These interactions induce hetero-aggregation of nanoplastics according to the surrounding media and the environmental context. The final hetero-aggregate form, size, and composition drive the final behaviour and impact as much as their size classes and aggregation behavior is linked to the environmental features (pH, salinity, sunlight, etc.). Hetero-aggregation, in which two or more different types of particles form aggregates, is more likely to occur than homoaggregation for nanoplastics due to the much higher number of natural colloids.

However, homoaggregation studied at the laboratory scale is still very useful to characterize the stability of nanoplastics as a function of their physicochemical properties and the salinity of the medium. At the ppm range of concentrations, the critical coagulation concentration (CCC) can be determined by dynamic or static light scattering and UV/visible spectroscopy. These methods have been used in recent years to study the stability of nanoplastics, and of all the existing methods [START_REF] Holthoff | Coagulation Rate Measurements of Colloidal Particles by Simultaneous Static and Dynamic Light Scattering[END_REF], the time-resolved DLS is the most common.

Table 1 summarizes recent laboratory studies investigating homoaggregation rates and CCCs of anionic-modified nanoplastics based on PS, PET, and PE. In general, the data show that the particle-particle attachment efficiency increases with the ionic strength; this is due to the compression of the electrical double layer, in agreement with the Deryaguin-Landau-Verwey-Overbeek (DLVO) theory. The importance of valence is shown in Table 1, where divalent cations (Ca 2+ ) destabilize nanoplastics at lower concentrations than monovalent cations (Na + ), following the Schulze-Hardy Rule (CCC ∝ 1/z n , with 2<n<6) [START_REF] Oncsik | Aggregation of Negatively Charged Colloidal Particles in the Presence of Multivalent Cations[END_REF].

At the same pH and for an equivalent size, the large spread of the CCC data can be explained by the nanoplastics' commercial origin. Manufacturers mainly use surfactants and are often unwilling to inform the end user even that surfactants were employed and how much remained in the latex dispersion, or about details of the surface charge density.

Nonionic surfactants attach covalently to the nanoparticle surface, and neither extensive dialysis nor ion exchange resins will remove them. There is usually significant batch-tobatch variability and differences between manufacturers, as well, which inevitably leads to differences in CCC experimental data.

Compared with commercial nanoplastics, those obtained from laser ablation or mechanical grinding are polydisperse with irregular shapes and complex surface chemistry, and they are free from residual surfactants. Thus, they are more suitable model materials to study the aggregation of nanoplastics in the environment. Recently, Veclin et al. employed nanoplastics free of surfactant and chemical control of the surface to model material surface weathering [START_REF] Cloé | Effect of the surface hydrophobicity-morphology-functionality of nanoplastics on their homo-aggregation in seawater[END_REF]. The samples presented a wide distribution of relevant surface properties such as functionality (ionizable carboxylic group, 0.10 to 1.7 mmol g -1 ), hydrophobicity (surface energy, 2.20 to 37.5 mJm-2), surface morphology (smooth or "raspberrytextured"), zeta potential (-31 to -21 mV) and anisotropy in shape. The CCC measurements demonstrate that spherical nanoplastics are more stable in seawater (CCC > 600 mmol L -1 ) than anisotropic nanoplastics (CCC ~ 100 mmol L -1 ) are. 50 280-1300 --3-10 UV-irradiation 24h [71] PET-laser 110 700 u U.V. /Vis Ablation laser [START_REF] Magrì | Laser Ablation as a Versatile Tool To Mimic Polyethylene Terephthalate Nanoplastic Pollutants: Characterization and Toxicology Assessment[END_REF] PET-grinded 300-700

401

The size, shape, and surface characteristics of any nanoplastic model material will affect the heteroaggregation phenomenon and, therefore, its stability in the environment.

As the first test of heteroaggregation, Cai et al. used fluorescent 100 nm commercial polystyrene standards (PSLs) with CaCl2, NaCl, and FeCl2 in the presence of humic substances [START_REF] Cai | Effects of inorganic ions and natural organic matter on the aggregation of nanoplastics[END_REF]. They demonstrated that with an inorganic heterogeneity such as Fe(III) in the hetero-aggregation, humic substances play the role of a stabilizing agent of PSLs.

Orikhova et al. came to the same conclusion when studying commercial 20 nm PSL-NH2 in the presence of Fe2O3 and alginate as model organic matter. Due to the amidine function, the ratio of the concentration of natural organic matter over PSL-NH2 affects the stability of the hetero-aggregates in fresh water [START_REF] Oriekhova | Heteroaggregation of nanoplastic particles in the presence of inorganic colloids and natural organic matter[END_REF]. The amidine group was also used by Wu et al.

using PSL-NH2 in humic acid to investigate the formation of hetero-aggregate and their impact on Daphnia magna [START_REF] Wu | Effect of salinity and humic acid on the aggregation and toxicity of polystyrene nanoplastics with different functional groups and charges[END_REF]. These authors demonstrated that the positively charged nanoparticles of polystyrene induced more toxicity that was tempered by the presence of humic acid. Tallec et al. came to another conclusion, which was that humic acid did not affect the aggregation and stability of PSL-NH2 compared to PSL-COOH as observed by Orikhova [START_REF] Oriekhova | Heteroaggregation of nanoplastic particles in the presence of inorganic colloids and natural organic matter[END_REF][START_REF] Tallec | Surface functionalization determines behavior of nanoplastic solutions in model aquatic environments[END_REF]. This can be explained by the presence of Tween20® residue in the standards used by Tallec et al. As previously described, the presence of an undeclared surfactant or another stabilizing agent in commercial nanoplastic model materials, such as PSLs, induces a change in the hetero-aggregation formation as observed for toxicological studies [START_REF] Pikuda | Toxicity Assessments of Micro-and Nanoplastics Can Be Confounded by Preservatives in Commercial Formulations[END_REF].

For negatively charged nanoplastics, the aggregation of PS, PET, or PE nanoplastics was significantly reduced with organic matter, and the CCC value increased substantially in NaCl solution. Pradel et al. showed that two types of natural organic matter (humic acid and alginate) enhanced the aqueous stability of anisotropic nanoplastics over time at high ionic strengths. Depending on the organic matter's nature, different stabilizing mechanisms were revealed using dynamic light scattering and asymmetrical flow field flow fractionation coupled to static light scattering [START_REF] Pradel | Stabilization of Fragmental Polystyrene Nanoplastic by Natural Organic Matter: Insight into Mechanisms[END_REF].

Humic acid provides electrostatic repulsion between particles, and some larger humic acid molecules causes a steric hindrance. Sodium alginate sorbs onto and separates particles and small aggregates of nanoplastics. The covered particles are stabilized by steric hindrance.

However, in divalent electrolytes, Yu et al. described the enhanced stability of PSL-COOH at low concentrations of Suwannee river natural organic matters (SRNOM) and altered stability at high concentrations of SRNOM due to the interparticle bridging effect caused by the Ca 2+ and carboxyl groups of SRNOM [START_REF] Yu | Aggregation kinetics of different surface-modified polystyrene nanoparticles in monovalent and divalent electrolytes[END_REF]. The cation bridging effect was also observed for anisotropic nanoplastics (PS-laser) in the presence of high concentrations of divalent electrolytes and SRNOM [START_REF] Yu | Aggregation kinetics of different surface-modified polystyrene nanoparticles in monovalent and divalent electrolytes[END_REF]. Similar results were shown by Singh et al., who noted faster aggregation with CaCl2 due to complexation with the dissolved organic matter [START_REF] Singh | Understanding the stability of nanoplastics in aqueous environments: Effect of ionic strength, temperature, dissolved organic matter, clay, and heavy metals[END_REF]. Otherwise, the mineral clay colloids participate in hetero-aggregation with nanoplastics under the influence of salts. The stability of nanoplastic colloidal suspensions decreased in the presence of the positively charged goethite or magnetite, while it was affected by the negatively charged montmorillonite and kaolinite, suggesting that there was a strong electrostatic attraction between nanoplastics and the two iron oxides [START_REF] Zhang | Charge mediated interaction of polystyrene nanoplastic (PSNP) with minerals in aqueous phase[END_REF]. To complexify the solution chemistry, Li et al. have shown that the hetero-aggregation of nanoplastics with metal-based nanoparticles (e.g., CeO2-nanoparticle of 25 nm) and humic acid was influenced by the nanoplastic surface chemical functionality (COOH, SO3H, C2H2O, NH2), the pH, and the presence of a divalent electrolyte (CaCl2) [START_REF] Li | Impact of CeO2 nanoparticles on the aggregation kinetics and stability of polystyrene nanoplastics: Importance of surface functionalization and solution chemistry[END_REF].

Some authors recently started to consider more relevant organic matter for marine systems, such as exopolymeric substances (EPS) from bacteria in the marine system. These EPS are polysaccharide macromolecules and present structures susceptible to interacting preferentially with nanoplastics through hydrophobic attraction even if the PSL used had surfactant traces [START_REF] Feng | Role of extracellular polymeric substances in the acute inhibition of activated sludge by polystyrene nanoparticles[END_REF]. Grassi et al. have analyzed the dynamic formation of an eco-corona coming from the interaction of PSL-COOH with EPS produced by the marine diatom P.

tricornutum [START_REF] Grassi | Interplay between extracellular polymeric substances (EPS) from a marine diatom and model nanoplastic through eco-corona formation[END_REF]. The chemical characterization revealed a predominance of polysaccharide material concerning proteins over a broad molecular weight range. The addition of EPS significantly reduces PSL-COOH aggregation in seawater [START_REF] Grassi | Interplay between extracellular polymeric substances (EPS) from a marine diatom and model nanoplastic through eco-corona formation[END_REF].

The authors of this review attempted to consolidate the studies on the hetero-aggregation of nanoplastics, and to investigate the surface influence on the hetero-aggregation formation. However, bias and lack of coherence on parameters such as 1) the functional group, 2) the size and shape of the model material, 3) the composition, and 4) the evaluation of the hetero-aggregation of nanoplastics made it difficult to complete our review. The hetero-aggregation and the final structure of the aggregates is a crucial piece to understand, as it will define nanoplastics' fate and impact. The influence of the functional group of the model material on the hetero-aggregation formation comes first. In the environment, the colloidal stability of the nanoplastics is governed by the organic matter sorbed and/or associated with it. Therefore, the surface charge and functionality does not influence its reactivity with natural media due to the corona of the natural organic matter and its specificities.

While it is intuitively accepted that nanoplastics result from the degradation of plastic fragments in seawater or on beaches, it is also possible that this degradation takes place in rivers [START_REF] Besseling | Fate of nano-and microplastic in freshwater systems: A modeling study[END_REF]. However, surface and groundwater are not a continuous phase and are made up of several systems that can transport, trap or re-suspend nanoparticles. These phenomena lead to a crucial question: what is the impact of this transport on the behaviour of nanoplastics? As described above, physicochemical studies of colloids show that nanoplastic aggregation depends on the ionic strength of the dispersion medium. These experiments do not consider the spatiotemporal dynamics of the medium's physicochemical properties, including the flux and ionic strength gradients. However, the particles, when in fresh surface water, will transit through gradients of significant ionic strength before reaching ocean water bodies, such as estuaries, lagoons, and mangroves [START_REF] Venel | Environmental Fate Modeling of Nanoplastics in a Salinity Gradient Using a Lab-on-a-Chip: Where Does the Nanoscale Fraction of Plastic Debris Accumulate?[END_REF].

Likewise, most ecotoxicological studies have not considered the complexity of natural aquatic environments either. The procedure is similar from one study to another: the PSLs (often at high concentrations) are dispersed in MilliQ water. This dispersion is then spread to the experimental exposure medium to obtain a given final concentration while handling a minimum sample volume. While this procedure makes sense for liquid contaminants, it is inappropriate for nanoplastics. The particles concentrated in freshwater are then suddenly re-dispersed in a medium with a distinct ionic strength. It should be added that the experimental conditions are favorable to aggregation phenomena, and the observed effects may be mistakenly attributed to nanoplastics while they are due to their agglomerates. As a result, it is possible that ecotoxicology studies underestimate, or poorly estimate, the effects of nanoplastics.

Conclusion

The scientific challenge that remains is to determine the behaviour of nanoplastics in aquatic environments, at the freshwater-saltwater and liquid-ice interfaces, which are essential areas for biodiversity. The other challenge will be to understand this environmental behaviour, and the mechanisms of action, and to determine whether nanoplastics are toxic.

However, due to the lack of feasible technologies, nanoplastics have not been yet sampled or quantified in the environment and the research needs material models. It should be mentioned that just as a single nanoplastic particle does not exist, a universal nanoplastic model material couldn't be claimed. However, a material become a relevant model as soon as its properties are properly controlled, known and reported. As shown, within influent properties, colloidal behavior is of previous importance since it involves not only the stability of the nanoplastic but also will help to understand its bioavailability.

Figure 1 .

 1 Figure 1. Total surface and concentration were calculated from different fractions of accumulated plastic waste (200Mt) within our oceans (V = 1400 10 6 km 3 ). The red dotted line is calculated from field data extrapolated by Lenz et al. using a power-law regression (n = 3.188*.d -2.67 ; with y as the particle concentration and x the particle size). The orange straight line is the surface of the earth (357.10 6 km 2 )

Table 1

 1 reports the main characteristics of the different kinds of nanoplastics.

Table 1 .

 1 Main characteristics of nanoplastic categories as defined as being below 1 µm. PSLs and nanoplastic model materials list the most common features reported within the literature. Real nanoplastic characteristics are given from the current knowledge and perspective. (*from this article list)

	Designation	Polystyrene latex (PSL)	Nanoplastic model materials	Real nanoplastics
	Origin	Commercially available	Bottom-up approach	Top-down approach	To be sampled
	Description	Polymer standard	From dispersed media	From primary plastic fragmentation)	From secondary plastic fragmentation	Environmental sample
	Morphology	spherical	spherical	Flakes, irregularly shaped particles	flakes, irregularly shaped particles	flakes, irregularly shaped particles
	Size dispersity	monodisperse	monodisperse	Continuum of size	Continuum of size	Continuum of size
	Polymer nature	PS	PS	polyolefins, PS	Polyolefins, PS	Potentially any kind of plastics
	Additive formula and content	From producer data	Known and controlled	From producer data	To be analyzed and determined	To be analyzed and determined
	Ad/Absorbed pollutants	N/A	N/A	N/A	To be examined and determined	To be analyzed and determined
	Surface functionality	Anionic, cationic, or none (qualitative information)	possibly known and adjustable	A brand new plastic	Potentially as oxidized plastic (aged debris)	Oxidized plastic (aged debris)
	Embedded tracers	Fluorescent, metal	Fluorescent, metal	Fluorescent, metal	N/A	N/A
	Key references*	and over 200 references within the literature	[50,53,55,61,6 2]	[58-60,63,64]	[60]	[22-24]

Table 2 .

 2 Review of recent studies reporting the homoaggregation rates and critical coagulation

	394
	395	concentrations (CCCs) in mono-and divalent electrolytes of anionic surface-functionalized nanoplastic
	396	(not chemically specified but with negative zeta potential, carboxylated, sulfonic and formic. The origin of
	397	the nanoplastics is commercial (emulsion), from laser ablation, mechanical fragmentation, U.V.
	398	irradiation, and filtration methods. * given with the plastic nature (PS, PE, PET) and the surface chemical
	399	function (none, COOH, SO3H, C2H2O) for the commercial nanoplastics (PSLs) or the elaboration methods,
	400	namely grinded, U.V., laser, filt.
			Size	CCC (mmol/L)
		nanoplastic*				pH	Method	Origin	Ref
			(nm)	NaCl	CaCl2
			240	140	25	6			[70]
				450	33	6			[71]
			100	300-500 460	--32	3 to 10 6	DLS	commercial	[71] [72]
		PSL		591 135	71 --	7.5 6			[73] [74]
				500	40	5 to 9	SLS	commercial +	[75]
			93 55	310 950	29 --	u 6	DLS	commercial	[76] [74]
			50	264	29	5			[77]
			110	--	10	u		commercial	[78]
		PSL-COOH	97	308	28	u	DSL	chemical surface	[76]
			50	191	16	5		commercial	[77]
			28	800	10	5			[79]
		PSL-SO3H	90	159	12	6	DLS	commercial	[80]
			80	264	29	5			[77]
		PSL-C2H2O	50	84	10	5	DLS	commercial	[77]
		PS-grinded	350	59	--	6.5	DLS	mechanical	[81]
				67		8		degradation	[81]
			100	957-1108	--	u		UV-irradiation in	[73]
								H2O2 -60 to 120h	[73]
		PS-UV	50-	760-458	27-8	6	DLS	UV-irradiation 1 to	[71]
			100					24h

Acknowledgements S.R. and B.G. thank the financial support of E2S (Energy and Environment Solutions) of the Universite de Pau et Pays de l'Adour. S.R. and A.A. acknowledge the support provided by the ANR through the PLASTI-SCARE project N°ANR-19-CE04-0007.

Conflicts of Interest

There are no conflicts of interest to declare.