Tathagata Mondal

Laurence Charles

Jean-François Lutz
email: jflutz@unistra.fr

Damages and Repair in Informational Poly(N-substituted urethane)s

The degradation and repair of uniform sequence-defined poly(N-substituted urethanes) was studied. Polymers containing an ω-OH end-group and only ethyl carbamate main-chain repeat units rapidly degrade in NaOH solution via an ω→α depolymerization mechanism with no apparent sign of random chain cleavage. The degradation mechanism is not notably affected by the nature of the sidechain N-substituents and took place for all studied sequences. On the other hand, depolymerization is significantly influenced by the molecular structure of the main-chain repeat units. For instance, hexyl carbamate main-chain motifs block unzipping and can therefore be used to control the degradation of specific sequence sections. Interestingly, the partially-degraded polymers can also be repaired; for example using a combination of N,N'-disuccinimidyl carbonate with a secondary amine building-block.

Overall, these findings open up interesting new avenues for chain-healing and sequence editing.

Informational polymers are macromolecules that store information in a controlled comonomer sequence. [1] A typical example of that is DNA, which is a biological storage medium. In recent years, it has also been reported that data (e.g. in a binary format) can be stored in synthetic macromolecules, for example in polyamides, polyesters and polyurethanes, just to name a few. [2] This new class of functional polymers has opened up interesting applications in areas such as data storage, cryptography, anti-counterfeiting, traceability and plastic recycling. [3] Yet, there are still pronounced differences between natural and non-natural informational polymers. Man-made polymers mainly enable passive information storage, while the information sequences of DNA can be copied, mutated, damaged, modified (i.e. through epigenetics) and repaired. [4] There are currently only a couple of examples of synthetic information sequences that can be degraded or transformed; [2c, 5] although such behaviors could be useful for the aforementioned applications as well as for more distant goals such as the development of artificial Life. [6] Here, we report the controllable degradation and possible repair of informational poly(N-substituted urethane)s. Sequence-defined polyurethanes can be synthesized via different solid-phase chemistry approaches. [2d, 7] Different types of functional sequences can be prepared, including digitally-encoded ones. [2d, 7e] It is also well-known that some linear or dendritic polyurethanes undergo controllable depolymerization (sometimes referred to as self-immolative behavior). [8] In these macromolecules, chain-degradation is triggered by a nucleophile (e.g. a primary amine) and proceeds via an unzipping mechanism. This interesting mechanism has been explored for biomedical applications, for instance for triggered drug release. [9] Very recently, Anslyn and coworkers [10] have reported that the depolymerization of sequence-defined polyurethanes can be kinetically-controlled to enable sequencing. [11] In this case, chain unzipping was triggered by a terminal OH-group and required basic conditions and elevated temperatures (70°C, microwave) to proceed. In the present work, we investigated the depolymerization in basic medium of sequence-coded poly(N-substituted urethane)s, a promising new family of digital polymers. [7e] As described in the following paragraphs, depolymerization occurs at room temperature and can be finely tuned by macromolecular design.

The poly(N-substituted urethane)s studied in this work were all synthesized using a recentlyreported orthogonal solid-phase synthesis protocol. [7e] Figure 1 shows their general structure. These informational polymers are directional and contains therefore two distinct chain ends denoted as α and ω. The former one is a caproic acid residue, whereas the latter one is either an OH or a methyl group.

The coded sequences were constructed using a set of nine different informational monomers A-I.

Figure S1 shows the molecular structure of the polymers P1-P13 studied in this work. The molecular uniformity of the polymers was assessed by high-resolution electrospray mass spectrometry (ESI-HRMS) and size exclusion chromatography (SEC), while their coded sequences were characterized by tandem mass spectrometry (MS/MS) (Table S1 and Figures S2-S14). [12] Figure 1. General structure of the informational poly(N-substituted urethane)s.

Since the depolymerization of sequence-defined poly(N-substituted urethane)s was never investigated, the degradation mechanism and kinetics were first analyzed in details. For polymers having an ωhydroxyl end-group and ethyl carbamate repeat units (x = 1 in Figure 1), complete depolymerization occurs at room temperature in the presence of sodium hydroxide. For example, Figure S2 shows the ESI-HRMS spectrum and SEC chromatogram of P1 having the sequence α-DCEC-OH after 16 hours in basic medium. Both analytical methods evidence the disappearance of P1 and the formation of three low molecular weight species of mass 129.1, 142.1 and 177.1 Da, which correspond to the

(A) R = Me, x = 1 (B) R = Et, x = 1 (C) R = Pr, x = 1 (D) R = Bu, x = 1 (E) R = Bz, x = 1 Y = OH or CH 3 α Sequence ω (F) R = Bz, x = 2 (G) R = Bz, x = 3 (H) R = Bz, x = 4 (I) R = Bz, x = 5
oxazolidinone rings resulting from the cyclization of repeat units C, D and E, respectively. As proposed in Scheme S1, these compounds are likely formed through a backbiting reaction leading to complete chain depolymerization. To validate this mechanism, P2 having the same primary structure as P1 but no terminal OH group (i.e. α-DCEC-CH3) was studied in the same basic conditions (Figure S3). This experiment indicated no sign of degradation, thus suggesting that depolymerization solely occurs through an ω→α unzipping mechanism, i.e. without contribution from other mechanisms such as random chain cleavage. [13] Furthermore, this behavior is fully orthogonal as shown in Figure 2, which compares the ESI-HRMS spectra of a mixture of P1 and P2 before and after basic treatment.

The selective depolymerization of P1 is observed in these conditions. Altogether, these results suggest a gradual unzipping degradation, as already reported for regular sequence-defined polyurethanes. [10] S5). P3 and P4 have a similar terminal unit but a different penultimate unit. Depolymerization proceeded in all cases and appeared to be sequence-independent and side-group-independent (for both penultimate and terminal positions). Yet, minor deviations in depolymerization kinetics cannot be ruled out but could not be evidenced in the performed experiments.

Furthermore, the influence of the number of the methylene groups in the repeat units was investigated. A series of polymers P4-P8 having all the same sequence but different terminal monomer units (R = Bz, x ranging from 1 to 5, Figure 1) was studied (Figures S5-S9). After 16h of basic treatment in methanol/water, the polymer having an ethyl terminal (P4) unit was entirely degraded.

The polymer having a propyl-based (P5) terminal units was significantly degraded but a residual peak of the initial polymer was still observed. For polymers having butyl-(P6) and pentyl-based (P7)

terminal units, the pristine polymer was still observed as a dominant species, although traces of degradation were detected. The polymer having a hexyl-based terminal unit (P8) obviously did not degrade in the studied conditions and only exhibited marginal oxazolidinone peaks. Behaviors comparable to those described for P5-P8 were obtained with polymers with alternative monomer sequences P9-P12. Altogether, these results imply that the depolymerization mechanism proceeds well when 5-and 6-membered rings are formed but is slowed down or hindered by repeat units leading to the formation of larger rings. To confirm this hypothesis, a mixture of polymers P4-P8 was treated in basic medium for 16h. ESI-HRMS analysis (Figure S17) indicated complete disappearance of P4, significant disappearance of P5, partial disappearance of P6 and P7, whereas P8 remained the most abundant species. Although these results should be interpreted with caution due to potential suppression effects during the ionization process, this experiment confirms that main-chain alkyls strongly influence the depolymerization mechanism. most likely due to the formation of polyesters, as discussed in previous publications. [7e, 14] The fact that chain depolymerization depends on the number of methylene groups in a repeat unit is appealing because it suggests that the content of an informational sequence can be tuned by monomer design. Indeed, in informational polymers, molecular bits are usually set by using side-chain motifs of different molar mass but can also be obtained using main-chain motifs of different length or composition. [15] For instance, in the present case, informational units containing hexyl spacers, such as monomer I, can be used as a stopper to interrupt depolymerization at a precise chain-location. As a proof-of-concept, polymer P13 with sequence α-DDBIAACE-OH was investigated (Figure 4a). It contains a single I unit placed in between two blocks that are composed of ethyl carbamate repeat units.

In NaOH medium, this polymer depolymerizes only until the stopping point, as evidenced by the ESI-MS spectrum (Figure 4b and Figure S18) and SEC chromatogram (Figure 4e) of the resulting polymer P13'. In ESI-MS, the transformation P13→P13' leads to the expected mass decrease of 508 Da, whereas in SEC an apparent diminution of 466 Da is measured between peak maxima Mp. Partial depolymerization is also supported by the appearance of the oxazolidinone rings, expelled from the decomposed AACE sequence, in the ESI-MS spectrum of P13' (Figure 4d). These results confirm that preset sequence sections can be selectively degraded using appropriate molecular design. Furthermore, the degraded chain still exhibits reactive chain ends (i.e. α-COOH and ω-OH) and can therefore be repaired. Hence, the same sequence (complete or partial) or a different one can be reconstructed on the partially-degraded polymer. Here, P13' was partially-repaired by reacting it first with N,N'disuccinimidyl carbonate (DSC) and then with di-n-propyl-amine. This modification leads to modification of both chain-ends, as shown in Figure 4c. Yet, it should be noted that the amidification of the α-chain-end can be avoided if needed; for instance by modifying the terminal acid group during or after cleavage. Here, the targeted modification was successfully performed, as evidenced by the ESI-MS spectrum (Figure 4c and Figure S19) and SEC chromatogram (Figure 4e) of the repaired polymer P13". Indeed, the transformation P13'→P13" led to the expected mass increase of 210 Da in ESI-MS and to an apparent Mp increase of 168 Da in SEC.

In summary, OH-terminated poly(N-substituted urethane)s degrade via an ω→α unzipping mechanism in basic conditions at room temperature. If the polymer chain is only composed of ethyl carbamate repeat units, full decomposition can be attained. Yet, if some repeat units of the chain contain longer alkyl spacers, degradation can be slowed down or even suppressed. For instance, polymers containing a hexyl carbamate terminal unit do not degrade in basic conditions; at least not notably during the investigated periods of times. Furthermore, if the hexyl carbamate unit is placed inside an otherwise ethyl carbamate-based sequence, partial depolymerization proceeds from the ωterminus to the stopping hexyl carbamate unit. Therefore, specific information sections can be selectively erased from an informational sequence. This property is particularly interesting for anticounterfeiting applications, [3b] because molecular barcodes could be purposely altered when exposed to specific conditions. Perhaps even more importantly, the partial degradation mechanism found in this work opens up possibilities of repairing informational polymers. For instance, it was demonstrated herein that intentionally-damaged sequences can be re-engineered using simple modification conditions. This proof-of-concept of molecular healing paves the way for the development of more complex deconstruction/reconstruction mechanisms, including chain rewriting and editing.

voltage: +75 V). Air was used as the nebulizing gas (10 psi) and nitrogen as the curtain gas (20 psi). Sample solutions were introduced in the ESI source with a syringe pump at a 10 μL min -1 flow rate. Ions were accurately mass measured after internal calibration of the orthogonal acceleration time-of-flight (oa-TOF) mass analyzer, using two cationic adducts of PMMA to bracket the targeted analyte m/z value in the MS mode [3] or the precursor ion signal in the MS/MS mode. In CID experiments, precursor ions were selected in a quadrupole mass analyzer, injected into the collision cell filled with nitrogen, and product ions were measured in the oa-TOF.

Instrument control, data acquisition and data processing of all experiments were achieved using Analyst software (QS 2.0) provided by Applied Biosystems SCIEX.

MS/MS sequencing of poly(N-substituted urethane)s has recently be described in details. [4] Briefly, collisional activation of ammonium adducts, [Px+NH4] + , first leads to elimination of ammonia to produce protonated molecules, [Px+H] + , which further experience cleavage of the CH2-O bond between two repeating units, yielding αi + fragments. A specific feature in these MS/MS spectra is an abundant internal fragment corresponding to he first repeating unit holding one proton. However, the abundance of αi + fragments rapidly decreases as the size oft he dissociating chain increases, so such a dissociation pathway allows sequencing of very short oligomers only (DP < 6). Alternatively, MS/MS sequencing can be efficiently performed from CID data recorded for sodiated oligomers, [Px+Na] + , using three fragment series, αi + ions containing the α termination and wi + and yi + ions containing the  termination. A fourth series of ions containing  and named w'i + are formed when activating doubly charged precursors,

[Px+2Na] 2+ . recorded before treatment in basic medium (inset: dissociation scheme for fragment assignment).

S5

C. Additional data and figures

(d) SEC chromatograms recorded in THF before (black trace) and after (blue trace) treatment in basic medium. The high molecular weight shoulder observed in the initial chromatogram might be due to polyester formation, as discussed in previous publications. [2,5] S16 recorded before treatment in basic medium (inset: dissociation scheme for fragment assignment).

(d) SEC chromatograms recorded in THF before (black trace) and after (blue trace) treatment in basic medium. The high molecular weight shoulder observed in the initial chromatogram might be due to polyester formation, as discussed in previous publications. [2,5] S18 recorded before treatment in basic medium (inset: dissociation scheme for fragment assignment).

(d) SEC chromatograms recorded in THF before (black trace) and after (blue trace) treatment in basic medium. The high molecular weight shoulder observed in the initial chromatogram might be due to polyester formation, as discussed in previous publications. [2,5] S20 recorded before treatment in basic medium (inset: dissociation scheme for fragment assignment).

(d) SEC chromatograms recorded in THF before (black trace) and after (blue trace) treatment in basic medium. The high molecular weight shoulder observed in the initial chromatogram might be due to polyester formation, as discussed in previous publications. [2,5] S22 (d) SEC chromatograms recorded in THF before (black trace) and after (blue trace) treatment in basic medium. The high molecular weight shoulder observed in the initial chromatogram might be due to polyester formation, as discussed in previous publications. [2,5] S24 treatment in basic medium. The high molecular weight shoulder observed in the initial chromatogram might be due to polyester formation, as discussed in previous publications. [2,5] S26 treatment in basic medium. The high molecular weight shoulder observed in the initial chromatogram might be due to polyester formation, as discussed in previous publications. [2,5] S28 treatment in basic medium. The high molecular weight shoulder observed in the initial chromatogram might be due to polyester formation, as discussed in previous publications. [2,5] S30 treatment in basic medium. The high molecular weight shoulder observed in the initial chromatogram might be due to polyester formation, as discussed in previous publications. [2,5] S32

Figure 2 .Figure 3 ,

 23 Figure 2. Positive mode ESI-HRMS spectra of a 1:1 w/w mixture of polymers P1 and P2: (a) before basic treatment and (b) after 16h at RT in NaOH solution in MeOH/H2O 2:1 v/v. The inset of panel b shows a zoom of the 120-180 m/z region of the obtained spectrum. As compared to the main spectrum,

Figure 3 .

 3 Figure 3. Evolution of SEC peak areas ratios as a function of time for the degradation of P1 in NaOH solution in MeOH/H2O 2:1 v/v. Peak areas were estimated by deconvoluting the chromatograms of Figure S15b using the Origin software. For a given time, the ratios α-DCE-OH/P1, α-DC-OH/P1 and α-D-OH/P1 are calculated from the corresponding chromatogram.

Figure 4 .

 4 Figure 4. Partial degradation and repair of sequence-coded poly(N-substituted urethane) P13. Positive mode ESI-HRMS spectra of (a) the original polymer P13 before basic treatment, (b, c) the degraded polymer P13' after NaOH treatment, and (d) the repaired polymer P13" after molecular healing. Panel c shows a zoom of the 100-200 m/z range for P13'. As compared to the spectrum shown in panel b, the intensity scale of the inset in panel c has been increased by a factor of about 11. The asterisks indicate background noise peaks. Experimental conditions: (i) NaOH, MeOH/H2O, RT (ii) DSC, pyridine, ACN, 60°C, microwave then di-n-propyl-amine, pyridine, DMF, RT. (e) SEC chromatograms recorded in THF for P13, P13' and P13". The high molecular weight shoulders are

Figure S5 .

 S5 Figure S5. Characterization of P4 (α-BCDE-OH). Positive mode ESI-HRMS recorded (a) before and (b) after treatment in basic medium, with with degradation products at m/z 116.1 (from B), m/z 130.1 (from C), m/z 144.1 (from D) and m/z 178.1 (from E) shown in inset. Peaks designated by an asterisk in (a) correspond to impurities. As compared to the inset, the intensity scale of the main spectrum shown in (b) was increased by a factor of about 7.

Figure S6 .

 S6 Figure S6. Characterization of P5 (α-BCDF-OH). Positive mode ESI-HRMS recorded (a) before and (b) after treatment in basic medium, with degradation products at m/z 116.1 (from B), m/z 130.1 (from C), m/z 144.1 (from D) and m/z 192.1 (from F) shown in inset. Peaks designated by an asterisk in (a) correspond to impurities. As compared to the inset, the intensity scale of the main spectrum shown in (b) was increased by a factor of about 35.

Figure S7 .

 S7 Figure S7. Characterization of P6 (α-BCDG-OH). Positive mode ESI-HRMS recorded (a) before and (b) after treatment in basic medium, with degradation products at m/z 116.1 (from B), m/z 130.1 (from C) and m/z 144.1 (from D) shown in inset. As compared to the main spectrum shown in (b), the intensity scale in the inset was increased by a factor of about 165.

Figure S8 .

 S8 Figure S8. Characterization of P7 (α-BCDH-OH). Positive mode ESI-HRMS recorded (a) before and (b) after treatment in basic medium, with degradation products at m/z 116.1 (from B), m/z 130.1 (from C) and m/z 144.1 (from D) shown in inset. Peaks designated by an asterisk in (a) correspond to impurities. As compared to the main spectrum shown in (b), the intensity scale in the inset was increased by a factor of about 9.

Figure S9 .

 S9 Figure S9. Characterization of P8 (α-BCDI-OH). Positive mode ESI-HRMS recorded (a) before and (b) after treatment in basic medium, with degradation products at m/z 116.1 (from B), m/z 130.1 (from C) and m/z 144.1 (from D) shown in inset. As compared to the main spectrum shown in (b), the intensity scale in the inset was increased by a factor of about 12.

Figure S10 .

 S10 Figure S10. Characterization of P9 (α-DDBF-OH). Positive mode ESI-HRMS recorded (a) before and (b) after treatment in basic medium, with degradation products at m/z 116.1 (from B), m/z 144.1 (from D) and m/z 192.1 (from F) shown in inset. Peaks designated by an asterisk in (a) correspond to impurities. As compared to the inset, the intensity scale of the main spectrum shown in (b) was increased by a factor of about 4.

Figure S11 .

 S11 Figure S11. Characterization of P10 (α-CCDG-OH). Positive mode ESI-HRMS recorded (a) before and (b) after treatment in basic medium, with degradation products at m/z 130.1 (from C), and m/z 144.1 (from D) shown in inset. Peaks designated by an asterisk in (a) correspond to impurities. As compared to the main spectrum shown in (b), the intensity scale in the inset was increased by a factor of about 65.

Figure S12 .

 S12 Figure S12. Characterization of P11 (α-DCAH-OH). Positive mode ESI-HRMS recorded (a) before and (b) after treatment in basic medium, with degradation products at m/z 102.1 (from A), m/z 130.1 (from C), and m/z 144.1 (from D) shown in inset. Peaks designated by an asterisk in (a) correspond to impurities. As compared to the main spectrum shown in (b), the intensity scale in the inset was increased by a factor of about 5.

Figure S13 .

 S13 Figure S13. Characterization of P12 (α-CACI-OH). Positive mode ESI-HRMS recorded (a) before and (b) after treatment in basic medium, with degradation products at m/z 102.1 (from A), and m/z 130.1 (from C) shown in inset. Peaks designated by an asterisk in (a) correspond to impurities. As compared to the main spectrum shown in (b), the intensity scale in the inset was increased by a factor of about 4.

Figure S14 .Figure S15 .Figure S16 .Figure S17 .

 S14S15S16S17 Figure S14. MS/MS recorded for P13 before treatment in basic medium, activated as [P13+2Na] 2+ at m/z 660.4. Top: dissociation scheme for fragment assignment. Peaks annotated by grey symbols are secondary fragments, including sodiated repeating units designated by stars: [A+Na] + at m/z 124.0, [B+Na] + at m/z 138.1, [C+Na] + at m/z 152.1, [D+Na] + at m/z 166.1, and [I+Na] + at m/z 256.1.

Figure S19 .

 S19 Figure S19. MS/MS recorded for P13" before treatment in basic medium, activated as [P13"+Na] + at m/z 999.7. Top: dissociation scheme for fragment assignment. Grey symbols designate secondary fragments, with @ for those arising from dissociation of the α end-group.

Table S1 .

 S1 Characterization of the poly(N-substituted urethane)s studied in this work.

		Sequence a	m/zexp	m/zth	Yield
					[%]
	P1	α-DCEC-OH	728.4470 b	728.4440
	P2	α-DCEC-CH3	726.4654 b	726.4648
	P3	α-CAAE-OH	658.3663 b	658.3658
	P4	α-BCDE-OH	714.4277 b	714.4284
	P5	α-BCDF-OH	728.4447 b	728.4440
	P6	α-BCDG-OH	742.4594 b	742.4597
	P7	α-BCDH-OH	756.4765 b	756.4753
	P8	α-BCDI-OH	770.4908 b	770.4910
	P9	α-DDBF-OH	742.4591 b	742.4597
	P10	α-CCDG-OH	756.4756 b	756.4753
	P11	α-DCAH-OH	742.4598 b	742.4597
	P12	α-CACI-OH	742.4592 b	742.4597
	P13	α-DDBIAACE-OH	646.8839 c	646.8836

a Sequence of the polymers written following the nomenclature rules established in Figure

1

of the main text. b Detected as [M+NH4] + in positive mode ESI-HRMS. c Detected as [M+H+NH4] 2+ in positive mode ESI-HRMS. d Measured by SEC in THF using a polystyrene calibration.

Acknowledgements

This work was supported by CNRS, the University of Strasbourg and the LabEx CSC. The SEC results shown in the publication were obtained with the help of the polymer characterization service of the Institut Charles Sadron. L.C. acknowledges support from Spectropole, the Analytical Facility of Aix-Marseille University, by allowing a special access to the instruments purchased with European Funding (FEDER OBJ2142-3341).

Supporting Information for

Damages and Repair in Informational Poly(N-substituted urethane)s

Tathagata Mondal, 1 Laurence Charles 2 and Jean-François Lutz* 1 The Wang resin (0.94 mmol/g, IRIS Biotech) utilized for the synthesis of Nsubstituted polyurethanes was modified with a cleavable linker as previously reported. [1] All the following reagents were purchased and used as received: 2-(methylamino)ethanol (TCI; ≥98%), 2-(ethylamino)ethanol (TCI; ≥98%), 2-(propylamino)ethanol (Sigma-Aldrich; ≥98%), 2-(butylamino)ethanol (TCI; ≥98%),di-n-propylamine (Alfa-aesar; 99%), 2-benzyl amino ethanol (TCI; ≥95%), 3-benzyl amino 1-propanol (TCI; ≥98%), 4-benzyl amino 1-butanol (TCI; ≥96%), 5-benzyl amino 1-pentanol (TCI; ≥98%), 6-benzyl amino 1-hexanol (TCI; ≥97%), trifluoroacetic acid (TFA ; Alfa-aesar, 99%), anhydrous pyridine (Sigma-Aldrich, 99.8%), N,N-disuccinimidyl carbonate (DSC; TCI, >98.0%), anhydrous acetonitrile (dry ACN; Sigma-Aldrich, 99.8%), dichloromethane (DCM, Sigma-Aldrich, 99.9%), diethyl ether (Carlo Erba), anhydrous N,Ndimethylformamide (dry DMF; Sigma-Aldrich, 99.8%), DMF (Sigma-Aldrich, R99.0%).

A.2. General procedure for the synthesis of poly(N-substituted urethanes) P1-P13. All the polymers P1-P13 were synthesized using a previously-reported stepwise solid phase protocol. [2] In brief, the synthesis is performed on an OH-functionalized modified Wang resin and involves two successive coupling steps. In the first step, the OH-groups of the resin are reacted with N,Ndisuccinimidyl carbonate in a microwave reactor (Monowave 300, Anton Paar). This step is performed in anhydrous acetonitrile and anhydrous pyridine at 60°C under microwave irradiation.

In the second step, the formed activated carbonate is reacted with the secondary amine function of an amino alcohol building block in a solid-phase extraction tube. This step is performed at RT in anhydrous DMF. Both steps were repeated a certain number of times until a desired chain-length and sequence were achieved. The cleavage of the polymers from the resin was achieved using a TFA/CH2Cl2 mixture. After filtering-off the resin, the organic layer with TFA was evaporated under reduced pressure to achieve the desired oligo/polyurethanes as pale yellow sticky solids.

A.3. Example of basic degradation of a poly(N-substituted urethane).

The poly(N-substituted urethane) P1 (25 mg, 0.035 mmol) was introduced in a small vial. Then 1.5 equivalent of NaOH (2 mg) in 2 mL of methanol and 1 mL of water (methanol/water 2/1 v:v) was added to the vial and the reaction mixture was stirred for 16 h at room temperature. After the alkaline treatment the reaction was stopped and methanol was evaporated out followed by the addition of an equivalent amount of aqueous HCl solution with respect to NaOH. Then the aqueous phase was extracted with dicholoromethane and the organic layer was collected, concentrated and dried under vaccum.

A similar procedure was applied for all polymers P1-P13.

S3

A.4. General procedure for repair of partially-degraded poly(N-substituted urethanes). The poly(N-substituted urethane) P13 (1 Eq., 50 mg, 0.039 mmol) was introduced in a small vial and reacted with NaOH following the procedure described in section A.3, thus affording the partiallydegraded polymer P13' as well as oxazolidinone rings resulting from the degradation. The cyclic residues were washed out by dissolving them in diethyl ether. Afterwards, P13' was treated with DSC (6 Eq, 60 mg, 0.234 mmol) in presence of anhydrous pyridine (10 Eq, 31 mg, 0.39 mmol) in 2 mL of anhydrous acetonitrile solution (60 o C, microwave, 1h, 600 rpm). Afterwards, an excess of di-n-propyl amine (20 Eq, 78 mg, 0.78 mmol) and 10 equivalent anhydrous pyridine (10 Eq, 31 mg, 0.39 mmol) along with 1 mL anhydrous DMF were added in the same reaction vial and stirred for 30 min. After the reaction, the solvents were evaporated under vacuum. Then, 2 mL acidic aqueous solution (1 M HCl solution) was added to the reaction mixture and extracted with dicholoromethane. The organic layer was separated and concentrated. Finally, the oily organic compound was washed with n-hexane to afford the sticky yellowish final product P13" with a yield of about 95%.

B. Characterization.

B.1. 1 H Nuclear Magnetic Resonance (NMR).

The polymers were analyzed by 1 H NMR spectrometry on a Bruker Avance 400 MHz spectrometer equipped with Ultrashield magnet. The spectra were recorded in deuterated chloroform (CDCl3) (Sigma-Aldrich; 99,8%).

B.2. Size Exclusion Chromatography (SEC).

The molecular weight and the distribution of the polymers were determined by size exclusion chromatography in THF solvent. The SEC instrument was equipped with four PLGel Mixed C columns (5 mm, 30 cm, diameter: 7.5 (d) SEC chromatograms recorded in THF before (black trace) and after (blue trace) treatment in basic medium. The high molecular weight shoulder observed in the initial chromatogram might be due to polyester formation, as discussed in previous publications. [2,5] S10 (d) SEC chromatograms recorded in THF before (black trace) and after (blue trace) treatment in basic medium. The high molecular weight shoulder observed in the initial chromatogram might be due to polyester formation, as discussed in previous publications. [2,5]


M Souzy

H Lhuissier

Y Méheust

T Le Borgne

B Metzger
email: bloen.metzger@univ-amu.fr

Velocity distributions, dispersion and stretching in three-dimensional porous media

Keywords: mixing and dispersion, porous media, chaotic advection

Velocity distributions, dispersion and stretching in three-dimensional porous media

Introduction

Dispersion and mixing processes in porous media are relevant to a broad range of fields and applications, such as hydrology, for contaminant transport in soils and aquifers [START_REF] Dentz | Mixing in groundwater[END_REF], the health industry and biology, for drug delivery and nutrient transport in brain or plant tissues (Nicholson 2015;[START_REF] Jensen | Sap flow and sugar transport in plants[END_REF][START_REF] Peyrounette | Multiscale modelling of blood flow in cerebral microcirculation: details at capillary scale control accuracy at the level of the cortex[END_REF], or the chemical and energy industries, where heat exchangers, filters and catalytic processes also rely on transport through porous media. Modelling these systems requires an understanding of how a tortuous assembly of interconnected channels or pores affects the dispersion and subsequent mixing of a given scalar (e.g. dye concentration or temperature) that is transported by a liquid flowing through them. These transport processes are fundamentally controlled by 891 A16-2 M. Souzy and others the distribution of pore-scale velocities. Because of the opaque nature of porous media, the characterization of these velocity fields is particularly challenging in three-dimensional (3-D) porous media.

Based on the velocity probability density function (p.d.f.), the dispersion process may be modelled using a continuous-time random walk (CTRW) [START_REF] Bijeljic | Predictions of non-Fickian solute transport in different classes of porous media using direct simulation on pore-scale images[END_REF][START_REF] Kang | Pore-scale intermittent velocity structure underpinning anomalous transport through 3D porous media[END_REF][START_REF] Holzner | Intermittent Lagrangian velocities and accelerations in three-dimensional porous medium flow[END_REF][START_REF] Dentz | Mechanisms of dispersion in a porous medium[END_REF]. In this framework, knowledge of the p.d.f.s in the low-velocity range is particularly important, since it determines the possible anomalous nature of transport in porous media [START_REF] Berkowitz | Modeling non-Fickian transport in geological formations as a continuous time random walk[END_REF]. Previous experimental studies based on optical index matching of bead-packs [START_REF] Morales | Stochastic dynamics of intermittent pore-scale particle motion in three-dimensional porous media: experiments and theory[END_REF][START_REF] Carrel | Pore-scale hydrodynamics in a progressively bioclogged three-dimensional porous medium: 3D particle tracking experiments and stochastic transport modeling[END_REF]) have reported increasing power-law distributions in the low-velocity range. However, numerical studies solving the Stokes equation in a random, isotropic, bead-pack with a pressure drop imposed at the boundaries (AlAdwani 2017; [START_REF] Dentz | Mechanisms of dispersion in a porous medium[END_REF]) have reported a flat distribution at low velocities. Therefore, although the dispersion of advective particles has been widely investigated in porous media, uncertainties in the velocity p.d.f. imply that active debate still remains about its nature. For non-diffusive particles, anomalous transport has been observed and modelled [START_REF] Kang | Pore-scale intermittent velocity structure underpinning anomalous transport through 3D porous media[END_REF][START_REF] Holzner | Intermittent Lagrangian velocities and accelerations in three-dimensional porous medium flow[END_REF] using a CTRW based on broad velocity distributions characterized by a decreasing power law at low velocity (stretched exponential). In such a case, a persistent anomalous dispersion is expected as a result of the large probability of low velocities. However, more recent works [START_REF] Bijeljic | Predictions of non-Fickian solute transport in different classes of porous media using direct simulation on pore-scale images[END_REF][START_REF] Dentz | Mechanisms of dispersion in a porous medium[END_REF] have shown that the presence of a minimal velocity or the introduction of diffusive motion [START_REF] Saffman | A theory of dispersion in a porous medium[END_REF]) makes the dispersion process transit to a Fickian regime. The persistent or transient nature of the anomalous dispersion observed in porous media should thus be clarified based on a more robust knowledge of the underlying advection field.

Moreover, velocity gradients that develop at the pore scale control the stretching of solutes and therefore their mixing rates [START_REF] Ottino | The Kinematics of Mixing: Stretching, Chaos, and Transport[END_REF][START_REF] Hinch | Mixing: turbulence and chaos an introduction[END_REF][START_REF] Aref | Frontiers of chaotic advection[END_REF]. Recent developments, in the so-called lamellar representation [START_REF] Ranz | Applications of a stretch model diffusion, and reaction in laminar and turbulent flows[END_REF][START_REF] Antonsen | The role of chaotic orbits in the determination of power spectra of passive scalars[END_REF][START_REF] Meunier | How vortices mix[END_REF][START_REF] Haynes | What controls the decay of passive scalars in smooth flows?[END_REF][START_REF] Duplat | A nonsequential turbulent mixing process[END_REF][START_REF] Le Borgne | The lamellar description of mixing in porous media[END_REF][START_REF] Villermaux | Mixing versus stirring[END_REF] have shown that knowledge of the stretching laws allows a quantitative prediction of mixing rates in complex flows. As an initially segregated blob of dye is advected in a complex flow and deforms into a set of elongated lamellae or sheets [START_REF] Ottino | The Kinematics of Mixing: Stretching, Chaos, and Transport[END_REF], the distribution of stretching rates provides the full statistics of elongations experienced by the blob. For simple flows, these stretching laws can be obtained analytically or using the diffusive strip method developed in both two and three dimensions [START_REF] Meunier | The diffusive strip method for scalar mixing in two dimensions[END_REF][START_REF] Martínez-Ruiz | The diffusive sheet method for scalar mixing[END_REF]. For more complex flows, [START_REF] Souzy | Stretching and mixing in sheared particulate suspensions[END_REF] have shown that the stretching laws can also be inferred from an experimentally measured velocity field. Recent theoretical and numerical investigations on bead-packs and idealized porous media have demonstrated that the flow topology in 3-D porous media must lead to exponential stretching [START_REF] Shaqfeh | Polymer stretch in dilute fixed beds of fibres or spheres[END_REF][START_REF] Lester | Is chaotic advection inherent to porous media flow?[END_REF]Lester, Dentz & Le Borgne 2016;[START_REF] Turuban | Space-group symmetries generate chaotic fluid advection in crystalline granular media[END_REF]. However, knowledge of the complete distribution of the material line elongations is still lacking. In a recent experimental study performed with a 3-D bead-pack [START_REF] Kree | Scalar mixtures in porous media[END_REF], an exponential mean stretching rate was found to provide good predictions for concentration statistics in the lamellar framework, where the mean and variance of stretching were derived as fitting parameters. However, in spite of their crucial importance for modelling how mixing proceeds, the distributions of stretching rates have never been directly measured at the pore scale in 3-D porous media.

Velocity, dispersion and stretching in 3-D porous media 891 A16-3

In the present study, we take advantage of recently developed experimental techniques -index matching [START_REF] Souzy | Stretching and mixing in sheared particulate suspensions[END_REF], the diffusive strip method [START_REF] Meunier | The diffusive strip method for scalar mixing in two dimensions[END_REF]) and the tracking algorithm (Heyman 2019) -to investigate the kinematics of dispersion and stretching in a 3-D random bead-pack. We provide here a highly resolved experimental reconstruction of the 3-D Eulerian fluid velocity field. Moreover, we developed a high-range velocity measurement technique to explore the shape of the velocity distribution at low velocities. The dispersion laws of non-diffusive tracers were measured and related to the fluid velocity distributions. Finally, we provide the first experimental characterization of the full distribution of stretching rates in a 3-D porous bead-pack. After presenting the experimental set-up in § 2, the 3-D velocity field and its characteristics are reported in § 3. The results about the dispersion are presented in § 4 and those about the stretching laws in § 5. Conclusions are drawn in § 6.

Experimental set-up

The experimental set-up is sketched in figure 1(a). The porous medium consists of a 3-D random stack of rigid, spherical, poly(methyl methacrylate) (PMMA) grains (Engineering Laboratories Inc.) of diameter d = 2 mm. The grains are tightly packed in a long transparent container (100d × 12d × 12d). The fluid is forced through the porous medium with a constant flow rate using a translating stage (M-403.4 PD, not shown in the schematics) and a syringe placed at the top of the set-up. The fluid is collected at the bottom in a flexible container.

Porous medium, liquid and tracers

The PMMA grains composing the porous medium have been specifically chosen for their smooth surface, good transparency and monodispersity. The solid volume fraction is close to the random packing fraction of frictional spheres 0.58 and the porosity (void fraction) is therefore 0.42. The fluid employed, Triton X-100, is Newtonian and has a large viscosity, η = 0.270 Pa s. The typical average fluid velocity is U ∼ 100 µm s -1 . This yields a Reynolds number ρ U d/η ∼ 10 -3 , ensuring that inertial effects are negligible. At room temperature, the fluid refractive index ( 1.492) is close to that of the PMMA grains ( 1.491), which makes the porous medium nearly transparent. To optimize the transparency, we finely match both indices of refraction down to the fourth digit [START_REF] Souzy | Stretching and mixing in sheared particulate suspensions[END_REF]) by adjusting the temperature of the water bath surrounding the porous medium to T = 28.1 ± 0.5 • C. The flow visualizations and the dispersion experiments have been performed by seeding the fluid with small fluorescent particles (PMMA B-particles from MF-Rhodamine) as shown in figure 1(b). The particles have a diameter d t = 3.23 µm, which is much smaller than the pore size (d t d/620). This allows an unprecedented resolution for passively sampling the fluid flow -as a comparison d t /d was around 1/63, 1/32 and 1/35 for the measurements by [START_REF] Moroni | Three dimensional particle tracking velocimetry studies of the transition from pore dispersion to Fickian dispersion for homogeneous porous media[END_REF], [START_REF] Datta | Spatial fluctuations of fluid velocities in flow through a three-dimensional porous medium[END_REF] and [START_REF] Carrel | Pore-scale hydrodynamics in a progressively bioclogged three-dimensional porous medium: 3D particle tracking experiments and stochastic transport modeling[END_REF], respectively. The particle diffusivity is negligible (the Péclet number based on their size, η U dd t /k B T, where k B is the Boltzmann constant, is larger than 10 10 ).

Imaging

The solid structure and fluid velocity inside the porous medium are imaged in a two-dimensional (2-D) plane formed by a planar vertical laser sheet, which is displaced across the porous medium to reconstruct the 3-D fields (see figure 1a). Each section of the porous medium illuminated by the sheet is imaged with a high-resolution camera (ORCA-Flash4.0, 2048 × 1048 pixels, 16 bit) mounted with a macroscopic lens (Sigma APO-Macro-180mm-F3.5-DG). This camera has been chosen for its linear response and its high signal-to-noise ratio. The laser sheet is formed by reflecting a laser beam (2 W, 532 nm) on a rotating mirror (∼10 4 r.p.m.), which yields a highly uniform sheet. It is focused down to a thickness of ≈60 µm using plano-convex lenses. Finally, scattered light is discarded with the help of a high-pass filter (590 nm) located before the lens; this ensures that only the light emitted by the fluorescent tracers is collected by the camera.

3. Three-dimensional velocity field 3.1. Velocity measurements Figure 1(b) shows a zoom on a typical image where the grain sections are highlighted in grey. All the grains have the same size. The apparent size differences only arise from their different distances to the laser sheet plane. The fluid velocity is measured by tracking the fluorescent tracers (bright dots in the image) using the open-source particle tracking velocimetry software TracTrac developed by [START_REF] Heyman | TracTrac: a fast multi-object tracking algorithm for motion estimation[END_REF]. The volume fraction of tracers is sufficiently small (∼10 -5 ) not to affect the flow. The algorithm tracks each individual tracer and provides its displacement between successive frames. Within this stationary flow, the velocity field is obtained by accumulating data over 1000 successive images (separated by 0.1 s) such that, in each pixel within the fluid, approximately 10 particles were detected and tracked. The strength of the present TracTrac algorithm is to provide the fluid velocity within each pixel (here the pixel size is d/228), thus largely outperforming any particle image velocimetry techniques in terms of spatial resolution. Figure 2(a) shows a map of the raw 2-D velocity amplitude U 2-D / U 2-D , with U 2-D ≡ √ u 2 + v 2 , obtained with the algorithm. The raw data are post-processed to successively (i) mask regions inside the grains, (ii) fill the few pixels where no tracer has been detected with the surrounding pixels' median value, and (iii) smooth the velocity field using a Gaussian The 3-D velocity field is obtained by repeating the above 2-D velocity field measurement over two perpendicular sets of 50 vertical sections spaced by d/10, as depicted in figure 1(a). The sampled volume of porous medium is 9d × 5d × 5d. The two sets of 2-D fields are linearly interpolated to provide the three velocity components (u, v, w) in the three directions (e x , e y , e z ) over the whole sampled domain (2048 × 1048 × 1048 voxels). Finally, a weighted divergence correction scheme [START_REF] Wang | Weighted divergence correction scheme and its fast implementation[END_REF]) is used to ensure that the interpolated field is divergence-free. Again, both the interpolation and the divergence-free algorithm have been checked to have a negligible effect on the velocity distributions.

The reconstructed 3-D velocity field is shown together with the porous medium microstructure in figure 3(a). It provides the three velocity components with a higher spatial resolution (d/228) than the most recent numerical simulations performed at the pore scale [START_REF] Maier | Simulation of flow through bead packs using the lattice Boltzmann method[END_REF][START_REF] Kang | Pore-scale intermittent velocity structure underpinning anomalous transport through 3D porous media[END_REF][START_REF] Jin | Statistics of highly heterogeneous flow fields confined to three-dimensional random porous media[END_REF].

Velocity distributions

The distribution of the velocity components (u, v, w) and magnitude U= √ u 2 + v 2 + w 2 are presented in figure 3(b-d). The velocity distributions are similar to those reported from previous experiments [START_REF] Datta | Spatial fluctuations of fluid velocities in flow through a three-dimensional porous medium[END_REF][START_REF] Holzner | Intermittent Lagrangian velocities and accelerations in three-dimensional porous medium flow[END_REF][START_REF] Carrel | Pore-scale hydrodynamics in a progressively bioclogged three-dimensional porous medium: 3D particle tracking experiments and stochastic transport modeling[END_REF] and numerical simulations [START_REF] Dentz | Mechanisms of dispersion in a porous medium[END_REF]. The distribution of the streamwise component (u) is skewed towards positive values and has an exponential tail. Small negative velocities, indicating the presence of back-flow, are present, although with a very weak probability. The distributions of the transverse velocities (v, w) are symmetric and have exponential shapes. Last, the distribution of the velocity magnitude is also exponentially tailed but it is found to follow, at low velocity, a power law, p.d.f. (U) ∝ U α , with a positive exponent, α 0.89. As discussed in the introduction, the characteristics of the velocity distribution at small velocities control the longitudinal dispersion behaviour [START_REF] Berkowitz | On characterization of anomalous dispersion in porous and fractured media[END_REF][START_REF] Bijeljic | Signature of non-Fickian solute transport in complex heterogeneous porous media[END_REF]. In order to clarify these crucial characteristics, we have refined the velocity measurements using a dynamic-range technique.

Refined measurements at low velocities

The tracking algorithm accurately detects the particle displacements from 0.1 to 5 pixels only. Below 0.1 pixel, the displacement signal compares with the noise level, and above 5 pixels, the algorithm is likely to swap neighbouring particles. This experimental limitation actually biases the velocity field, especially at low velocity. To circumvent this problem, a dynamic-range measurement is used. It consists in performing three measurements of the velocity field with a constant frame rate (as described in § 3.1) but with a 10-fold increase of the flow rate between each measurement. Note that this dynamic range measurement was performed only over a single vertical plane since the statistics of the fluid velocity over that plane is found to be identical and thus representative of that obtained over the whole sampled volume. For each measurement, only the velocities corresponding to particle displacements in the confidence range of 0.5-5 pixels are retained.

Figure 4(a) shows the regions corresponding to the three velocity ranges of measurement, with the fastest regions of the flow (measured with the slowest flow rate) in yellow, the intermediate regions in light blue and the slowest regions (largest flow rate) in dark blue. The unbiased distribution of the 2-D velocity magnitude obtained with this dynamic-range technique is found to be flat at low velocity (p.d.f.(U 2-D ) ∝ U 2-D 0 ); see figure 4(b). This refined measurement procedure was not extended to the 3-D velocity field because of time constraints. However, since both p.d.f.(u) and p.d.f.(v) are flat at low velocity (see appendix A), the fact that p

.d.f.(U 2-D ≡ √ u 2 + v 2
) is also flat indicates that u and v are strongly correlated for small velocities (small values of v are found where u is small). Moreover, by symmetry, w must be similarly correlated to u. Therefore, since the three velocity components are correlated and since their distribution is flat at low velocity, the 3-D velocity magnitude distribution, p.d.f.(U 3-D ≡ √ u 2 + v 2 + w 2 ), must be flat as well at low velocity, in agreement with the numerical simulations performed recently by AlAdwani (2017) and [START_REF] Dentz | Mechanisms of dispersion in a porous medium[END_REF].

Before discussing the implications of the shape of the velocity distribution on the dispersion process (see next section), we provide below an empirical correlation between the magnitude of the velocity and its orientation. 3.4. Velocity magnitude and orientation Figure 5 presents the statistics of the streamline local orientations p.d.f.(cos θ ) and the mean velocity magnitude for a given orientation U cos θ obtained from the 3-D experimental field over the velocity range for which measurements are accurate (U 0.3 U ). The streamline orientation is found to be well fitted by p.d.f.(cos θ ) ∝ e a cos θ , with a strong dependence on cos θ embedded in the value of a 4; see inset of figure 5(a). This trend simply reflects that streamlines are strongly aligned with the longitudinal direction (e x ). Besides, we find that the magnitude of the velocity is also strongly correlated with its orientation. The average velocity Velocity, dispersion and stretching in 3-D porous media 891 A16-9 magnitude for a given orientation U cos θ increases when streamlines are more aligned with the flow, following the exponential trend U cos θ ∝ e b cos θ , with b 1.5.

These two experimental observations suggest a compact, 3-D characterization of the one-point statistics of the velocity field. Assuming for simplicity that the relative distribution of the velocity for each orientation is independent of the orientation (which is presumably a reasonable ansatz given the isotropy of the medium structure) imposes that the joint distribution of U and θ follows

Anomalous dispersion?

4.1. Dispersion measurements The dispersion process induced by this flow field is characterized in two ways: (i) directly, by monitoring the evolution of a blob of small non-diffusive particles (see figure 6), and (ii) indirectly, by numerically advecting non-diffusive point particles in the experimental 3-D velocity field (see figure 7).

The first method consists in injecting with a thin needle a small blob of fluid, typically of the size of the grains, seeded with non-diffusive fluorescent tracers (see § 2) in the bulk of the porous medium. Once the injection is completed, the flow is turned on and the evolution of the blob is captured within one plane illuminated by the laser sheet (see figure 6). The dispersion process is quantified through image processing by measuring the spatial variance σ 2 of the tracer locations in both the longitudinal and transverse directions. The measurement can be achieved until t/τ = 10, since all tracers have to remain in the field of view. The experiment is repeated 13 times by injecting the blob at different locations within the porous medium.

The second method consists in using the 3-D experimental fluid velocity field to advect numerically non-diffusive point particles. Some examples of trajectories are provided in figure 7. A total of 1250 tracers are positioned randomly within the fluid phase and advected using an explicit Runge-Kutta scheme. In order to allow a longtime advection (t/τ > 100), each point particle that reaches a boundary of the fluid domain is re-injected upwards at a location where the fluid velocity is the closest (in both magnitude and orientation) to its current velocity. This re-injection procedure was found to have no effect on the dispersion statistics. As shown in figure 3(d), the velocity distribution of these Lagrangian point particles is identical to that of the Eulerian velocity field, as expected in the present case of a volume-conserving flow [START_REF] Dentz | Continuous time random walks for the evolution of Lagrangian velocities[END_REF].

The dispersion results are gathered in figure 8. dispersion envelope explored by these non-diffusive particles is indicated with a dotted line. Figure 8(b) shows the trajectories of the 1250 numerical point particles (in grey) as well as their positions at t/τ = 0, 43 and 136 (in colour). The dispersion process is quantified by measuring the variance of the tracers' position, in both the longitudinal and transverse directions, i.e. σ i 2 ≡ i 2 -i 2 , where i stands for x, y or z. The variances obtained with the two methods are compared in figure 8(c) until t/τ = 10. After a short ballistic regime, where σ 2 ∝ t 2 for t/τ 1, the variance becomes seemingly anomalous for 1 t/τ 10. It is super-diffusive in the streamwise direction (σ 2

x ∝ t 1.4 ) and sub-diffusive transversely (σ y 2 ∝ t 0.7 ). Note that similar exponents have also been reported for the same range of advection times by [START_REF] Kang | Pore-scale intermittent velocity structure underpinning anomalous transport through 3D porous media[END_REF], who simulated the fluid flow in the 3-D pore space of a real rock. Moreover, although the 3-D velocity field is slightly biased at small velocities (as discussed in § 3.3), the streamwise and transverse dispersions obtained from the numerical advection are very similar to those measured directly in the set-up, which validates the numerical advection procedure.

To characterize the dispersion over longer times, we rely on numerical advection measurements. These measurements, presented in figure 8(d), show that the anomalous trends are only transient. For t/τ 10, both the longitudinal and transverse dispersions eventually relax to a Fickian regime with variances following

Link between dispersion and velocity distributions

The above measurements require some interpretation. The ballistic dispersion regime lasts typically until t ∼ τ , which indicates that the velocity of the tracers remains correlated over a displacement of order d. Beyond that time, the dispersion becomes anomalous.

In the longitudinal direction, velocities are essentially positive and the (superdiffusive) anomaly is controlled by the occurrence of the smallest velocities, which significantly delay some of the tracers. Following [START_REF] Berkowitz | Modeling non-Fickian transport in geological formations as a continuous time random walk[END_REF], we model the displacement of each tracer as a succession of random longitudinal steps with a fixed length d and a random velocity u that is conserved over the step but uncorrelated with the previous velocities. This CTRW is fully constrained by the velocity distribution. It is computed numerically for 10 4 uncorrelated tracers (the first step length is chosen randomly between 0 and d, to account for the random initial position of the tracers, and the tracers' displacement is sampled with a substep scale at desired times). For an exponential velocity distribution, which has the same flat shape (p(u) ∝ u 0 ) at low velocity and the same mean value u / U = 0.79 as the experimental distribution, the CTRW yields σ 2

x /d 2 ∼ log(t/τ )t/τ . As shown in figure 9, this trend is in good agreement with the experimental data over the range 1 t/τ 10 and agrees with the prediction that the dispersion is super-diffusive when the low-velocity exponent of the distribution (p(u) ∝ u α ) satisfies α 0 [START_REF] Berkowitz | Modeling non-Fickian transport in geological formations as a continuous time random walk[END_REF]. To understand the final relaxation towards a Fickian regime, the finite minimal velocity of the tracers has to be considered. Indeed, in a real flow, advected tracers always have a finite minimal velocity. This finite minimal velocity can arise from the finite size d t of the tracers, which prevents them from sampling the smallest fluid velocities close to the walls. In the present case, this minimal velocity can be estimated as U min / U ≈ 10d t /d ≈ 0.02, since the typical pore size is equal to d/10; see figure 3(a). The finite minimal velocity can also result from the finite thermal diffusivity of the tracers. Such tracers can escape from a region of low velocity more quickly through the effect of molecular diffusion than through the effect of advection alone [START_REF] Saffman | A theory of dispersion in a porous medium[END_REF]. In such a case, the minimal velocity is U min ∼ (10 U /d) × δ, where δ ∼ √ Dd/U min is the transverse diffusive displacement and D is the tracers' (thermal) diffusion coefficient. This yields U min / U ≈ (10 2 D/ U d) 1/3 . This minimal velocity has a crucial impact on the longitudinal dispersion, precisely because the Velocity, dispersion and stretching in 3-D porous media 891 A16-13 x /d 2 and σ z 2 /d 2 , obtained from the numerical advection in the 3-D experimental velocity field compared to the CTRW simulations. CTRW based on a flat velocity distribution (p.d.f.(u) ∝ u 0 at small u) yields a persistent anomalous dispersion σ 2

x /d 2 ∼ log(t/τ )t/τ (green line), which matches the experiments until t/τ ∼ 10. For a flat distribution truncated below U min / U = 0.01, 0.03, 0.1 and 0.2, the dispersion eventually becomes Fickian (σ 2

x ∝ t) after t F /τ ≈ 100, 33, 10 and 5, respectively (grey lines). The experimental transverse dispersion is well captured by a CTRW with anticorrelated velocities (see text). The black dashed lines indicate the slope 1. dispersion anomaly is controlled by the low velocities. To illustrate this point, we simulate a CTRW based on the same exponential distribution but discarding all velocities below U min . As shown in figure 9, when U min / U is set to 0.01, 0.03, 0.1 or 0.2, the variance relaxes to a Fickian regime beyond t F /τ 100, 33, 10 and 5, respectively, instead of following log(t/τ )t/τ as previously. The Fickian regime is reached once the velocities of all the tracers become decorrelated. The CTRW dispersion matching the experiments is obtained with U min / U = 0.1. This is consistent with the fact that the 3-D velocity distribution, as a result of the limited experimental resolution, departs from a flat distribution below U/ U 0.1 (see figure 3d).

By contrast, the sub-diffusive anomaly in the transverse direction is not controlled by the smallest velocities but is a consequence of the meandering trajectories inside the porous medium, which makes the transverse velocities alternately positive and negative with similar magnitudes. This can be shown by simulating a unidimensional CTRW with a large probability p that the tracers change direction between steps. Using an exponential velocity distribution with a mean value v / U = 0.34 identical to the experimental distribution (figure 3c) and setting p = 0.86 results in a sub-diffusive dispersion in good agreement with the experiments. As shown in figure 9, the transient sub-diffusive trend is recovered over the typical duration τ /(1 -p), over which velocities remain anticorrelated.

With the above information at hand, we can explain the values of the dispersion coefficients (D and D ⊥ ) obtained in the (long-time) Fickian regime. For tracers with a constant velocity over the porous scale d, the duration of each step is τ = d/U. 891 A16-14

M. Souzy and others

The probability of τ is fully constrained by the tracer velocity distribution p U (U) according to p τ (τ ) = (d 2 / U τ 3 )p U (d/τ ), which is defined over 0 τ d/U min (see details in appendix B). Now, the probability that a tracer has moved over a large curvilinear distance s, i.e. that s/d steps have been covered at a given time t, is given by the s/d-fold convolute of p τ , i.e. p τ * s/d (t). Using the central limit theorem and x /s = u / U , the streamwise spatial variance σ 2

x can be expressed in terms of the variance of the step durations, σ τ

2 ≡ τ 2 -τ 2 , as σ 2 x /d 2 ≈ u 2 / U 2 σ τ 2 t/τ 3 .
For a flat velocity distribution (p U (U) ∝ U 0 ), the step duration variance is prescribed by the minimal velocity as σ τ 2 /τ 2 ≈ ln( U /U min ). Altogether, one obtains

D = σ 2 x 2t ≈ 1 2 ln U U min u 2 U 2 d 2 τ , (4.3)
which makes precise how the dispersion coefficient depends on the minimal velocity U min . With the experimental values u 0.79 U and U min 0.1 U , equation (4.3) yields D 0.7d 2 /τ , which is within the same order of magnitude as the measured value 0.45d 2 /τ reported in (4.2).

In the transverse direction, the same arguments together with the probability, 1 -p, that a step is not anticorrelated with the previous ones, give similarly

D ⊥ = σ y 2 2t = σ z 2 2t ≈ 1 2 ln U U min (1 -p) v 2 U 2 d 2 τ . (4.4)
For the experimental values v = w 0.34 U and U min 0.1 U , this yields D ⊥ 0.02d 2 /τ , which is also found to be close to the measurement 0.027d 2 /τ reported in (4.2).

Stretching laws

We now turn to the key information required to model mixing, namely, the statistics of the stretching of the fluid material lines.

Method

The stretching distributions are determined by advecting numerically material lines in the experimental velocity field. We follow the same procedure as in [START_REF] Souzy | Stretching and mixing in sheared particulate suspensions[END_REF], initially developed by [START_REF] Meunier | The diffusive strip method for scalar mixing in two dimensions[END_REF], but adapted here to a 3-D velocity field. The material lines consist of segments joining non-diffusive tracers initially separated by a distance l 0 = d/120. Each tracer is advected individually, but every time the length of a segment is doubled, the segment is split in two by adding a new tracer in the middle. The mean stretching rate, averaged over many segments, is found to be independent of the integration error. Moreover, by using different smoothing procedures, we checked that the residual noise arising from measurement artefacts and from the interpolation of the velocity field does not affect the value of the mean stretching rate. For instance, applying an additional smoothing procedure after the interpolation of the velocity field reduced the mean residual velocity divergence by 20 % but had no consequence on the value of the mean stretching rate, which remains constant within 1 %.

When advecting segments, we used the same re-injection procedure as for the dispersion measurements but applied to the segment centre. Again, we checked that this re-injection has no effect on the stretching statistics. The statistics of elongations are renormalized to account for the fact that the most elongated segments become more numerous in time. This procedure allows us to resolve the small-scale stretching and folding of the line at all times. An example of the evolution of a material line is where l(t) is the current line length between initially adjacent tracers.

Results

Figure 10(b,c) shows the experimental stretching laws obtained from the statistics of 6000 initial segments advected for a duration t/τ ≈ 20 (no initial transient is observed because the tracers have been pre-advected for t = 5τ to obtain a steady-state distribution of segment orientation). We find that both the mean logarithm of the elongation, ln ρ , and its variance, σ (5.

3)

The distribution of elongation is thus log-normal and its two first moments, i.e. the mean elongation, ρ , and the variance, σ ρ 2 , follow ρ = e (λ+µ/2)t/τ e (3/2)λt/τ and σ ρ 2 = e 2(λ+µ)t/τ e 4λt/τ .

(5.4a,b)

The exponential stretching revealed by the above advection procedure is reminiscent of Lagrangian chaos. The intensity of such chaotic dynamics is usually characterized by the Lyapunov exponent λ, which quantifies the average long-term growth rate of infinitesimal fluid element lengths. In our case, the Lyapunov exponent is simply the average steady-state stretching rate of the line, λ ≡ τ ρ/ρ t τ .

(5.5)

Figure 11(a) shows the evolution of τ ρ/ρ t τ as a function of time when no preadvection has been performed. The orientation of the segments is random at t/τ = 0. As expected from fluid incompressibility, the mean stretching rate is equal to zero initially; see inset of figure 11(a). Then, over of a few strain units, the segments orient with the flow and τ ρ/ρ converges to its steady-state value λ 0.47. We also estimate the anisotropy of the material line stretching by computing the mean components of the stretching rate ρ/ρ|e l • e x | , ρ/ρ|e l • e y | and ρ/ρ|e l • e z | , where e l ≡ l/|l| is the line tangent direction. Note that this definition does not include the effect of the line rotation; thus it only quantifies the line stretching. These stretching rates are thus different from the growth rate of the projected elongations. As shown in figure 11(b), each component follows a similar evolution, with an initial transient and a relaxation towards a positive steady-state value. Interestingly, we find that the mean stretching rate parallel to the flow, ρ/ρ|e l • e x | , is only 20 % larger than those measured in the transverse directions. This shows that, despite the symmetry breaking imposed by the mean flow, the random bead-pack structure of the porous medium induces strong fluctuations, which yield a close-to-isotropic stretching of fluid material lines.

Conclusion and discussions

Using advanced index-matching and particle-tracking techniques, we measured the pore-scale velocity in 3-D porous media at an unprecedented resolution. This allowed us to characterize in particular the low tail of the velocity p.d.f., which is critical for anomalous dispersion, and the distribution of local velocity gradients that control stretching and mixing. The three velocity components have been measured over a sample volume of 9d × 5d × 5d with a resolution of d/228. This extensive set of data, which may be used as a benchmark for numerical simulations, has been made available online [START_REF] Souzy | Mixing lamellae in a shear flow[END_REF], as well as the purposely developed 3-D velocity field explorer software Pore Aventura (Pore Aventura 2019).

Particular attention has been paid to the statistics of small velocities, which control the anomalous longitudinal dispersion trend frequently reported in porous media. Dynamic-range velocity measurements, increasing the accuracy at low velocity, have allowed us to show experimentally that the distribution of the velocity magnitude is flat at low velocity (p.d.f.(U) ∝ U 0 ), in agreement with recent numerical results (AlAdwani 2017; [START_REF] Dentz | Mechanisms of dispersion in a porous medium[END_REF].

The dispersion process has been investigated with non-diffusive tracers by three methods: (i) experiments with blobs of small particles; (ii) numerical advection on the experimental 3-D velocity field; and (iii) continuous-time random walk (CTRW) simulations. These investigations have shown that the dispersion process is transiently anomalous (σ 2

x /d 2 ∼ ln(t/τ )t/τ and σ y,z 2 /d 2 t/τ ) until a time scale t F ≈ d/U min set by the low-end cutoff U min of the tracer velocity distribution, before it eventually becomes Fickian (D 0.45d 2 /τ and D ⊥ 0.027d 2 /τ ) beyond t F . This low-velocity cutoff is always present owing to the finite size of the tracers (U min / U ≈ 10d t /d) or to their finite diffusivity (U min / U ≈ (10 2 D/ U d) 1/3 ). It also determines the value of the long-time longitudinal dispersion coefficient (D ∝ ln( U /U min )d 2 /τ ).

Finally, we provide the first experimental characterization of the full stretching statistics in a 3-D porous medium. The mean elongation of fluid material lines is found to grow exponentially in time ( ρ/ρ ≡ λ/τ ). This exponential trend is in sharp 891 A16-18 M. Souzy and others contrast to the linear growth reported from 2-D simulations and 2-D experiments performed at the pore scale [START_REF] De | The filamentary structure of mixing fronts and its control on reaction kinetics in porous media flows[END_REF][START_REF] Jiménez-Martínez | Impact of saturation on dispersion and mixing in porous media: photobleaching pulse injection experiments and shear enhanced mixing model[END_REF], or the power-law trends obtained for 2-D simulations at the Darcy scale (Le Borgne, [START_REF] Le Borgne | Stretching, coalescence, and mixing in porous media[END_REF][START_REF] Le Borgne | The lamellar description of mixing in porous media[END_REF], where the 2-D nature of these steady flows precludes the development of chaotic advection. Our experimental results thus confirm the chaotic nature of advection within 3-D porous media anticipated by [START_REF] Shaqfeh | Polymer stretch in dilute fixed beds of fibres or spheres[END_REF] and Lester et al. (2016).

The Lyapunov exponent value λ 0.47 measured here is larger than those reported in numerical studies. Lester et al. (2016) obtained λ 0.1178 for a model of random open network and [START_REF] Turuban | Space-group symmetries generate chaotic fluid advection in crystalline granular media[END_REF] found λ 0.073 for a crystalline sphere pack. These differences could stem from the specific model of random open network considered and from the crystalline nature of the sphere pack, which presumably induces less stretching than the random packing investigated here. This point remains to be understood.

Last, measurements of the full statistics of elongations revealed that the long-time distribution of elongations is log-normal. This suggests an underlying multiplicative stretching process where, at small scales, material lines undergo multiplicative and uncorrelated stretching cycles having a finite persistence time. This paradigm has been reported previously for other complex flows, such as a random sine flow [START_REF] Meunier | The diffusive strip method for scalar mixing in two dimensions[END_REF] or the flow within a sheared suspension of non-Brownian particles [START_REF] Souzy | Stretching and mixing in sheared particulate suspensions[END_REF]), and appears to be generic to a large class of complex flows.

The above results provide a solid basis for a full description of mixing within a porous medium. By characterizing the flow kinematics, the laws of dispersion and stretching provided here enable predictions for the evolution of the concentration distribution of a diffusive scalar [START_REF] Villermaux | Mixing versus stirring[END_REF]. They also appeal for further experimental work to obtain quantitative measurements of the evolution of the concentration distribution (e.g. using the techniques developed by [START_REF] Souzy | Mixing lamellae in a shear flow[END_REF]). Finally, it is worth recalling that these kinematic laws were obtained in a random pack of monodisperse spheres, that is to say, in an isotropic porous medium with a narrow pore size distribution. Whether they could serve as a kinematic framework for predicting mixing processes in real porous media encountered in geophysics, such as soils composed of sand or rocks, would be an interesting subject to address in future studies.

A16-20

M. Souzy and others made a large number s/d of steps, is simply the s/d-fold convolute of p τ , that is to say, p τ * s/d (t). From the central limit theorem and under the present assumption of a Fickian dispersion, the spatial variance σ s 2 ≡ s 2 -s 2 can be expressed in terms of the variance of the step durations, σ τ 2 ≡ τ 2 -τ 2 , according to

σ s 2 d 2 ≈ σ τ 2 τ 2 t τ . ( B 4)
In the long-time limit one has both x /s = u / U and x 2 /s 2 = u 2 / U 2 . Therefore, the streamwise variance follows Making use of (B 5) and (B 7), one recovers the expression provided in (4.3):

σ 2 x d 2 ≈ u 2 U 2
D = σ 2 x 2t ≈ 1 2 ln U U min u 2 U 2 d 2 τ , ( B 8)
which specifies how the dispersion coefficient depends on the minimal velocity U min .

In the transverse direction, the same arguments apply if one realizes that among n pairs of steps, only the (1 -p)n pairs for which the two steps have the same direction contribute to the dispersion. The variance is thus decreased by the same factor (1 -p), and replacing u by v in (B 8) yields the expression provided in (4.4): 9)

D ⊥ = σ y 2 2t = σ z 2 2t ≈ 1 2 ln U U min (1 -p) v 2 U 2 d 2 τ . ( B

  FIGURE 1. (a) Schematics of the set-up. (b) Detail of a typical sectional view of the porous medium. The grains are coloured in grey. The white dots are passive fluorescent tracers seeding the fluid. (c) Overlay of successive images illustrating the fluid flow.

  FIGURE 2. (a) Detail of a typical raw 2-D velocity field obtained by particle tracking. (b) Same field after post-processing. (c) Corresponding distributions of the 2-D velocity magnitude, U 2-D ≡ √ u 2 + v 2 .

  FIGURE 3. (a) Experimental 3-D velocity field and microstructure. The colour scale encodes the fluid velocity magnitude, U = √ u 2 + v 2 + w 2 . (b,c) Distributions of the velocity components (b) u along the mean flow direction (e x ) and (c) v, w along the transverse directions (e y , e z ). (d) Distribution of the velocity magnitude.

  FIGURE 4. Dynamic-range velocity measurements. (a) Map showing the regions of the three ranges of measurement: (i) yellow, fastest regions of the flow measured at low flow rate; (ii) light blue, intermediate regions; (iii) dark blue, slowest regions measured at large flow rate. (b) Comparison between the velocity distributions obtained with the dynamicrange technique and with a single measurement (same data as in figure 2c).

  FIGURE 5. (a) Distribution of the streamline orientation cos θ and average velocity magnitude for a given orientation U cos θ . Here A and B are normalizing constants (A = a/sinh a, B = 2(a + b) sinh a sinh b/[ab sinh(a + b)]). Inset: definition of the streamline orientation θ . (b) Relative distribution of the velocity magnitude for a given orientation. The black line represents (3.2). Inset: joint distribution of the magnitude and orientation.

  p.d.f.(U, cos θ) = 1 U cos θ p.d.f.(cos θ ) p.d.f.(U/ U cos θ ), (3.1) where, as shown in figure 5(b), the relative velocity distribution is found experimentally to be close to p.d.f.(U/ U cos θ ) = e -U/ U cos θ . (3.2) Equation (3.1) summarizes all the one-point information about the 3-D velocity field. It captures the main dependences observed experimentally, including an accurate description of the statistics at small velocity, since it embeds p.d.f.(U) ∝ U 0 for U U cos θ .

FIGURE 6 .

 6 FIGURE 6. Dispersion measurements: successive experimental images of a blob of small non-diffusive tracers flowing through the index-matched 3-D bead-pack (see also supplementary movie 1, available at https://doi.org/10.1017/jfm.2020.113). Only those particles contained in the z = 0 plane are visible (the field of view is 20d long in the streamwise direction and the tracers have a diameter d t = d/620).

FIGURE 7 .

 7 FIGURE 7. Dispersion measurements: trajectories of non-diffusive point particles advected numerically in the 3-D experimental velocity field (see also supplementary movies 2 and 3 for an immersive exploration of the 3-D velocity field (Pore Aventura 2019)). The six thick lines highlight six specific trajectories.

FIGURE 8 .

 8 FIGURE 8. Dispersion results. (a) Overlay of 13 blobs of non-diffusive tracers injected at 13 different locations in the porous medium at t/τ = 0 (red), 3 (green) and 10.5 (blue), with τ = d/ U . (b) Trajectories of 1250 non-diffusive point particles advected numerically in the experimental velocity field (grey) and their positions at t/τ = 0 (red), 43 (green) and 136 (blue). (c) Variances of the position σ i 2 ≡ i 2 -i 2 , with i = x, y or z, obtained with both methods. The error bars represent the typical standard deviation between experimental runs. (d) Same data over a longer time.

  FIGURE 10. (a) Evolution of a material line advected numerically in the 3-D experimental velocity field (see also supplementary movie 4). The line is displayed at four successive times. (b) Projection of the line on the transverse plane (e y , e z ) at the same times. (c) Mean and variance of the logarithm of the elongation versus time (σ 2 ln ρ ≡ ln ρ 2 -ln ρ 2 ). (d) Distribution of the logarithm of the elongation at four successive times. The black lines show the Gaussian distributions having the same mean and variance as the experimental ones.

  and others shown in figure10(see also supplementary movie 5). As the line is advected across the pores, it stretches and folds with patterns reminiscent of chaotic advection. We consider the local elongation of the line ρ

  2 ln ρ ≡ (ln ρ) 2 -ln ρ 2 , grow linearly in time and with almost identical slopes, according to ln ρ = λt/τ and σ 2 ln ρ = µt/τ , (5.2a,b) with λ µ 0.47 ± 0.03. Moreover, as shown in figure 10(c), the distribution of the elongation logarithm is well captured by the Gaussian distribution p.d.f.(x = ln ρ)

  FIGURE 11. (a) Mean stretching rate versus time. Inset: same data in semi-logarithmic scales. The dashed line shows the Lyapunov exponent (long-time value) λ 0.47 ± 0.03. The error bar reflects the time fluctuations of the Lyapunov exponent once steady state is reached. (b) Mean projected stretching rates along e x , e y and e z versus time, where e l ≡ l/|l| is the line tangent direction.

  anomalous dispersion condition: for a vanishing minimal velocity (U min = 0), the variances σ 2 τ and σ 2 x are finite only if the low-velocity exponent of the Lagrangian velocity distribution is positive (p U (U) ∝ U α with α > 0). For a finite U min and a flat velocity distribution (p U (U) ∝ U 0 ), the dependence to U min follows

https://www.cambridge.org/core/terms.

Acknowledgements

We warmly thank C. Wang for making his weighted divergence correction scheme available, J. Heyman for doing so with his particle tracking algorithm, and D. Lester, R. Turuban, E. Villermaux and P. Meunier for fruitful discussions. We also thank S. Noel for machining the experimental set-up, and N. Barret and L. Forner for developing Pore Aventura, a dedicated 3-D velocity field explorer software. This work has been supported by ANR ScienceFriction, by Labex MEC ANR-10-LABX-0092, by ANR-11-IDEX-0001-02 and by ERC consolidator grant ReactiveFronts 648377.

Declaration of interests

The authors report no conflict of interest.

Supplementary movies

Supplementary movies are available at https://doi.org/10.1017/jfm.2020.113.

FIGURE 12. Distribution of the velocity components u and v and of the 2-D velocity magnitude U 2-D = √ u 2 + v 2 obtained from the dynamic-range velocity measurement.

Appendix A. Refined distributions of the velocity components

Figure 12 shows that the velocity distributions of both components u and v, obtained with the dynamic-range measurement technique, are flat at low velocities.

Appendix B. Dispersion coefficients in the Fickian regime

We derive the expressions for the dispersion coefficients D and D ⊥ in the (long-time) Fickian regime for the CTRW considered in § 4.2, i.e. by modelling the displacement of tracers as a succession of random steps with a fixed length d and a random velocity U that is conserved over the step but uncorrelated with the previous velocities. In this framework, the probability for the duration, τ = d/U, of each step is fully constrained by the tracer velocity distribution. Indeed, the probability p τ (τ ) is directly related to the velocity distribution sampled per step (i.e. sampled per unit of length rather than per unit of time) p U|step according to

This distance-sampled velocity distribution is simply related to the Lagrangian (timesampled) distribution p U (U) that is actually measured, by

Together with (B 1) this gives

From the statistics of τ , those for the total distance travelled by a tracer in the limit of a large number of steps can be obtained as follows. The probability that, at given time t, a tracer has displaced by a large curvilinear distance s, i.e. that it has