ES2953864B2 - METHOD AND DEVICE FOR GASEOUS FOCALIZATION OF TRANSONIC FLUIDS OF WEAKLY VISCOELASTIC JETS - Google Patents

Description

DESCRIPTION

METHOD AND DEVICE FOR GAS FOCUSING OF FLUIDS

TRANSONICS OF WEAKLY VISCOELASTIC JETS

FIELD OF INVENTION

The present invention relates to a method of controlled production of small fluid entities. More specifically, the invention relates to a method of producing highly stable, weakly viscoelastic stationary jets by transonic gaseous fluid focusing.

FIELD OF INVENTION

Several techniques for the controlled production of small fluid entities (e.g., droplets, bubbles, emulsions, capsules) with multiple applications are known today. Gaseous fluid focusing (see A. M. Gañán-Calvo, Generation of steady liquid microthreads and micron-sized monodisperse sprays in gas streams, Phys. Rev. Lett. 80, 285-288 (1998)), is a paradigmatic example, where a liquid is injected at a constant flow rate through a supply capillary located in front of an orifice or inside a convergent nozzle. At the same time, a confluent gas stream is forced to cross the orifice/nozzle, originating favorable pressure gradients and viscous stresses, whose resulting forces sharpen the injected liquid forming a liquid meniscus attached to the supply capillary, and push a thin jet of liquid from the tip of said meniscus. When the liquid is injected with a flow rate above a threshold value Qmin and, additionally, the external gas stream is transonic (i.e., said stream has velocities equal to or greater than the speed of sound), it is possible to produce straight, very long and thin jets, with velocities from tens to around 100 m/s, and discharge them into a low-pressure chamber.

Gaseous fluid focusing has many important applications. The breakup of single or compound jets, due to capillary instability phenomena, allows the formation of small droplets/capsules with an acceptable degree of monodispersity. Furthermore, microbubbles can be ejected at the tip of the meniscus when the outer (focusing, continuous) and inner (focused, dispersed) phases are exchanged. In all these cases, the size of the microfluidic entity produced is smaller than any section of the ejector, due to the action of the focusing stream. Gaseous fluid focusing has also been used for the production of microparticles with complex structures or to form stimulus-sensitive microbubbles with a wide range of applications in pharmacy and medicine.

On the other hand, transonic gaseous fluid focusing has emerged as one of the most successful ways to introduce samples in serial femtosecond crystallography, or “Serial Femtosecond Crystallography” (SFX) (see H. N. Chapman et al., Femtosecond X-ray protein nanocrystallography, Nature 470, 73–77 (2011), and S. Boutet et al., High-Resolution Protein Structure Determination by Serial Femtosecond Crystallography, Science 337, 362–364 (2012)), which has revolutionized the molecular detection of complex biochemical species from proteins to viruses.

The above applications of gaseous fluid focusing require the production of very fine and long-lived jets. For example, in SFX, the longer the jet produced, the easier it is to avoid possible interferences (e.g., damage or splashing) between the X-ray radiation used in molecular detection and the jet originating the jet. Additionally, in order to reduce as much as possible the background noise present in the diffraction pattern obtained after irradiating the jet containing the sample in SFX, the diameters of the jets produced should be in the order of micro or nanometers. However, the diameter and length of the jets are commonly linked through the Weber and Ohnesorge numbers, so that, typically, the finer the jet, the shorter it is. Furthermore, in the pursuit of such an increase in relative length, the chemical composition of the liquids used to introduce the samples into SFX must not be significantly affected, so that the background noise in the pattern is known and/or easy to subtract.

It is therefore desirable to obtain a gaseous fluid focusing method that increases the stability of the jets, so that finer and longer jets are produced, and the chemical composition of the liquid that makes up the jet is not significantly modified.

The present invention solves the aforementioned needs.

BRIEF DESCRIPTION OF THE INVENTION

The present invention addresses the above-mentioned needs in the state of the art by means of a method that produces highly stable, weakly viscoelastic stationary jets by means of gaseous focusing of fluids.

In a first inventive aspect, the invention provides a method for gaseous fluid focusing of a working liquid, the method comprising the steps of providing a convergent zone comprising a receiving section and a discharge orifice, the convergent zone being configured to receive in the receiving section a gas stream and to discharge said gas stream downstream of the discharge orifice, wherein the ratio of the stagnation pressure (p0) in the convergent zone to the pressure (pc) downstream of the discharge orifice is at least that necessary for the velocity of the gas stream downstream of the discharge orifice to be equal to or greater than the speed of sound,

providing an inner capillary within the convergent zone, the inner capillary extending along an axis, the inner capillary having an inner surface with an inlet section and an outlet section perpendicular to the axis and also having an outer surface such that a meniscus-forming zone exists between the outlet section of the inner surface and the discharge orifice,

receive a constant flow of working liquid into the inlet section of the internal capillary,

form a meniscus of working liquid in the meniscus formation zone when the working liquid exits the inner capillary and contacts the gas stream, exerting a deformation rate on the working liquid, giving rise to the meniscus to a stationary jet of working liquid that is emitted from the meniscus to the discharge orifice.

Furthermore, the method is characterized in that the working liquid is a solution of a polymer with a molecular weight equal to or less than 5*106 g/mol, and more particularly, with a molecular weight equal to or less than 2*106 g/mol, and more particularly, with a molecular weight equal to or less than 1xio6 g/mol, where the concentration of the polymer in the solution is between a first value and a second value, where the first value is the concentration corresponding to a Weissenberg number equal to 0.5 in the meniscus formation zone, and where the second value is the concentration corresponding to a Weissenberg number equal to 10 in the meniscus formation zone.

The main advantage of this method is that the stationary jet of working liquid emitted from the meniscus to the discharge orifice is a highly stable, weakly viscoelastic jet. This stability is achieved both in the meniscus formation zone and downstream of the discharge orifice, and is much higher compared to the case where the working liquid was a Newtonian liquid. As a consequence of this improved stability, the weakly viscoelastic jet is considerably thinner at the discharge position, since lower working liquid flow rates are allowed to be ejected compared to Newtonian working liquids, whose stability limit in the meniscus formation zone is reached at higher flow rates. Furthermore, the improved stability is also manifested in that the weakly viscoelastic jet is considerably longer downstream of the discharge orifice compared to a Newtonian liquid jet.

Stability in the meniscus formation zone is related to the balance between the different stresses suffered by the working liquid in said meniscus formation zone. In transonic gas focusing of fluids, the working liquid is received with a constant flow rate in the inlet section of the internal capillary, and suffers an intense extensional flow near the meniscus surface, while said meniscus surface is dragged by the transonic gas stream, so that the speed of the working liquid is accelerated in the meniscus formation zone. Said increase in the speed of the working liquid takes place in distances of the order of tens or hundreds of microns, which is the typical length of the meniscus in transonic gas focusing of fluids, and entails a strain rate s' exerted on the working liquid in the meniscus formation zone.

Once the working liquid flow rate is above a minimum threshold, an equilibrium is established between the stresses suffered by the liquid (due to pressure gradients, tensional forces and viscous stresses), so that a stationary jet is emitted from the meniscus. If, in addition, the working liquid is a polymer solution, elastic stresses appear that contribute to the stabilization of the working liquid flow in the meniscus formation zone. In particular, a dilute polymer solution can behave as a fluid with low elasticity (characterized by a low relaxation time) and a practically constant viscosity under shear stresses. Therefore, the main effect of the polymer in such a solution is a slight increase in viscosity and elasticity, where elasticity is characterized by a low relaxation time, i.e., the relaxation time, which depends on the working liquid and the polymer concentration therein, will always be equal to or less than 200 ^s, commonly equal to or less than 100 ^s, frequently equal to or less than 50 ^s and sometimes equal to or less than 30 ^s.

On the other hand, given a polymer solution characterized by a relaxation time A* and subjected to an extensional flow that exerts on it a deformation rate s', it is possible to quantify the intensity of said extensional flow through a dimensionless quantity known as the Weissenberg number (Wi), which is given by Wi = Ar s'. Curiously, when this number is greater than 0.5, the dissolved polymers undergo a transition in which they go from being arranged in a random helical configuration to being in a stretched configuration, also called coil-stretch transition.

Likewise, the relaxation time Ar of a polymer solution increases as the polymer concentration in the solution increases, due to the hydrodynamic (i.e., flow-mediated) interaction between the polymers. Thus, for a given polymer solution, the strain rate s' required to trigger the helical-stretch transition of the polymers in a polymer solution decreases as the polymer concentration in the solution increases.

In this context, for a given strain rate s' exerted on the working liquid in the meniscus formation zone in transonic fluid gas focusing, if said working liquid is a dilute polymer solution whose viscosity is approximately constant under shear stresses, there exists a first value ci* for the polymer concentration in the polymer solution at which the Weissenberg number is equal to 0.5 in the meniscus formation zone.

Based on this, if in this context the concentration c of the polymer in the polymer solution is lower than said first value ci* (which we will call the viscoelastic concentration value), its respective Weissenberg number will be below 0.5 in the meniscus formation zone and, consequently, the helical-stretch transition will not take place, and the polymer solution will behave as a quasi-Newtonian liquid.

On the contrary, if said concentration c is equal to or greater than said ci* value, its respective Weissenberg number will be equal to or greater than 0.5 in the meniscus formation zone, so the helical-stretch transition will take place, and elastic stresses will be activated in the working liquid in the meniscus formation zone. As a consequence, said elastic stresses will help to stabilize the working liquid flow in the meniscus formation zone, and the stationary jet regime will be maintained for lower flow rates, compared to the case in which the working liquid used was a Newtonian liquid. And more importantly, since the flow rate value necessary to produce a stationary jet is lower, jets with a smaller diameter at the discharge orifice are obtained, compared to the case in which the working liquid used was a Newtonian liquid.

On the other hand, when the polymer concentration in the polymer solution exceeds a second value c2*, a pull-out effect occurs, caused by the imbalance that arises in the meniscus formation zone between the tension force originated by the Poiseuille-type flow in the inner capillary and that exerted by the emitted jet, so that the meniscus separates from the outlet section of the inner capillary. As a consequence, the triple contact line of the meniscus moves irregularly within the inner capillary, symmetry is lost and both the meniscus and the jet tend to suffer whip instabilities. This second value c2* (which we will call the pull-out concentration value) corresponds to a Weissenberg number equal to 10.

Likewise, given a dilute polymer solution, the extraction concentration value c2* will be greater than the viscoelastic concentration value c1* when the molecular weight Mw of the polymer of said solution is equal to or less than 5*106 g/mol, commonly equal to or less than 2*106 g/mol, and frequently equal to or less than i x i or 6 g/mol.

With respect to the improved stability of weakly viscoelastic jets downstream of the discharge orifice, compared with the case where the working liquid was a Newtonian liquid, such improved stability results in a noticeable delay in jet breakup. Specifically, such jet breakup occurs at distances which are several orders of magnitude larger than in the Newtonian case, contrary to expectations (see, for example, S. Middleman, Stability of a viscoelastic jet, Chem. Eng. Sci. 20, 1037-1040 (1965)).

In preferred embodiments of the invention, the working liquid is received in the inlet section of the internal capillary at a flow rate between 1.01 and 1.1 times the minimum flow rate compatible with the emission of a stationary jet of working liquid from the meniscus to the discharge orifice. The conditions for establishing the stationary jet are known to those skilled in the art, and relate to the formation of a stable meniscus from the tip of which a fine jet is emitted in a stationary manner, avoiding the appearance of dripping when the jet is emitted.

As a result, the diameter of the stationary jet at the discharge orifice is reduced to the minimum value. Furthermore, due to the elastic stresses that arise in the working liquid in the meniscus formation zone when the polymer concentration in the polymer solution is between the values c1* and c2*, the stability of the working liquid is improved compared to the case in which the working liquid was a Newtonian liquid. As a result, the value of the minimum flow rate required for the emission of a stationary jet and, consequently, the value of the minimum jet diameter at the discharge orifice decrease drastically compared to the case in which the working liquid used was a Newtonian liquid.

In other preferred embodiments of the invention, the polymer is at least one of polyethylene oxide (PEO), polyvinylpyrrolidone (PVP), fibroin, a short chain peptide or any polymer which, dissolved in water at a concentration of 1% by weight, has a relaxation time equal to or less than 200 s, and more particularly, a relaxation time equal to or less than 100 s, and more particularly, a relaxation time equal to or less than 50 s, and more particularly, a relaxation time equal to or less than 30 s. In this context, the weakly viscoelastic jet is obtained without significantly modifying the composition of the working liquid.

In other preferred embodiments of the invention, the gas stream is an inert gas stream. Examples of an inert gas stream are a stream of dry, impurity-free air, or a stream of helium, or a stream of nitrogen. This prevents polymer degradation in the dilute polymer solution used as the working liquid.

In other preferred embodiments of the invention, the pressure (pc) downstream of the discharge orifice is less than 0.1 bar, and more particularly less than 0.01 bar, and the stagnation pressure (p0) in the convergent zone is greater than 10 times the pressure (pc) downstream of the discharge orifice, and more particularly greater than 50 times the pressure (pc) downstream of the discharge orifice. This allows the velocity of the gas stream downstream of the discharge orifice to be equal to or greater than the speed of sound, maintaining a low pressure (pc) downstream of the discharge orifice, thus eliminating the need to maintain a high stagnation pressure (p0) in the convergent zone. A low pressure (pc) downstream of the discharge orifice is beneficial, since absolute whip instability phenomena are avoided. In addition, the working liquid experiences an intense extensional flow near the meniscus surface, which leads to a deformation rate in the meniscus formation zone compatible with the activation of the helical-stretch transition of the polymer solution. Furthermore, this intense extensional flow causes a delay in the relaxation of the elastic stresses downstream of the discharge orifice.

In other preferred embodiments of the invention, the discharge orifice has a diameter between 20 and 250 µm, and the separation between the outlet section of the inner surface of the inner capillary and the discharge orifice is between 0.5 and 5 times the diameter of the discharge orifice. This advantageously allows transonic velocities of the gas stream to be achieved with acceptable pressure ranges, such that the resulting intense extensional flow of working liquid in the meniscus formation zone gives rise to a deformation rate compatible with the activation of the helical-stretch transition of the polymer in the polymer solution. Furthermore, said intense extensional flow gives rise to a delay in the relaxation of the elastic stresses downstream of the discharge orifice.

In other preferred embodiments of the invention, the convergent zone comprises a flat plate portion, in which the discharge orifice is comprised, such that the flat plate portion extends perpendicularly to the axis of the inner capillary. More preferably, the convergent zone is provided open to the outside, such that the stagnation pressure (p0) in the convergent zone is the ambient pressure, while the region downstream of the discharge orifice comprises a receptacle comprising the flat plate portion and side walls which are arranged at a distance of at least 3 times the width of the discharge orifice, and more preferably at a distance of at least 5 times the width of the discharge orifice, such that the pressure (pc) downstream of the discharge orifice is regulated to achieve that the velocity of the gas stream downstream of the discharge orifice is equal to or greater than the speed of sound. As indicated above, regulating the pressure (pc) downstream of the discharge orifice more easily prevents the meniscus from suffering absolute whip instability and, consequently, expands the flow rate range compatible with the emission of a stationary jet of working liquid.

Another object of the present invention provides a method for performing serial femtosecond crystallography (SFX) comprising the steps of a method according to any of the previous embodiments, wherein the working liquid comprises a sample, and the method further comprises the steps of irradiating with pulses of X-ray radiation the stationary jet of the working liquid comprising the sample, and processing in a photodetector the signal resulting from the interaction between the pulses of X-ray radiation and the stationary jet of working liquid comprising the sample. Typically, in SFX the pulses of X-ray radiation are very short (from one to one hundred femtoseconds). Examples of samples comprised in the working liquid would be, among others, micro- or nano-crystals of biomolecules, large biomolecules, bio-reagents, viruses or parts of cells. In this context, since the breakup of the weakly viscoelastic jet is considerably postponed with respect to the case in which the working liquid comprises a sample, the method further comprises the steps of irradiating with pulses of X-ray radiation the stationary jet of the working liquid comprising the sample, and processing in a photodetector the signal resulting from the interaction between the pulses of X-ray radiation and the stationary jet of working liquid comprising the sample. Since the working liquid is a Newtonian liquid, a longer time frame and working length are allowed. This allows to easily avoid possible interferences (e.g., damage or splashes) between the X-ray radiation and the device producing the weakly viscoelastic jet. Furthermore, the thinness of the weakly viscoelastic jet allows to reduce the background noise present in the diffraction pattern obtained after irradiating the jet of working liquid containing the sample. Also, since the working liquid is a dilute polymer solution, the composition of the working liquid is not significantly affected, so that the background noise present in the diffraction pattern is well known and/or easy to subtract, and therefore the sample signal can be identified in said diffraction pattern.

Another object of the present invention provides a method for producing microparticles comprising the steps of a method according to any of the preceding embodiments, and additionally the steps of waiting for the stationary jet of working liquid to break up, and obtaining microparticles resulting from the break up of the stationary jet of working liquid. Typically, when the stationary jet of working liquid does not encounter an obstacle downstream of the discharge nozzle (e.g., the boundaries of a chamber delimiting the region downstream of the discharge nozzle), instabilities (e.g., whip instabilities, Beads-on-a-String instabilities) suffered by said jet downstream of the discharge nozzle cause it to eventually break up. Due to the considerable thinness of the weakly viscoelastic jet, the droplets/capsules resulting from the jet breakup are formed with a considerably reduced size, compared to the case in which the working liquid was a Newtonian liquid. In this context, the resulting droplets/capsules are suitable for producing microparticles with complex structures for a wide range of applications in pharmacy, biomedicine and even printed electronics.

Another object of the present invention provides a method for producing microbubbles comprising the steps of a method according to any of the preceding embodiments, and additionally the steps of

providing one or more additional capillaries, arranged within the inner capillary and concentric to said inner capillary, each additional capillary extending along the same axis, each additional capillary having an inner surface with an inlet section and an outlet section perpendicular to the axis and also having an outer surface,

receiving a constant flow rate of an additional working liquid at the inlet section of each additional capillary,

form microbubbles composed of each additional working liquid from each of the additional capillaries and the working liquid from the inner capillary, when each additional working liquid exits each additional capillary and contacts the working liquid exiting the inner capillary,

wait for the stationary jet of working liquid to break, and obtain the microbubbles after the stationary jet of working liquid breaks.

The use of additional concentric capillaries to obtain microbubbles is known in other fields of the art, as can be seen for example in document US 2003/098021 A1. Advantageously, this type of arrangement, combined with the conditions defined in the most general aspect of the present invention, allows obtaining much smaller microbubbles. Typically, these microbubbles are capsules formed by an outer fluid and an inner fluid suitable for a wide range of applications in pharmacy and biomedicine.

Another object of the present invention relates to the polymeric compound obtained from the stationary jet produced by a method according to any of the previous embodiments. Said polymeric compound advantageously presents a stretched configuration characterized by a relaxation time towards the equilibrium configuration, where said relaxation time is greater than the respective relaxation time characterizing the polymer of the polymeric solution originally received in the inlet section of the internal capillary. In obtaining said polymeric compound, the conditions experienced by the working liquid in the meniscus formation zone may give rise to a certain entanglement and/or self-assembly of the stretched polymers.

Another object of the present invention relates to a gaseous fluid focusing device suitable for carrying out the steps of a method according to an embodiment of the first inventive aspect, the device comprising

a discharge chamber delimiting a discharge region,

an ejector disposed within the discharge chamber, the ejector comprising a convergent zone comprising a receiving section and a discharge orifice open to the discharge region, the convergent zone being configured to receive in the receiving section a gas stream and to discharge said gas stream downstream of the discharge orifice,

where

The ejector further comprises an inner capillary disposed within the discharge zone, the inner capillary extending along an axis, the inner capillary having an inner surface with an inlet section configured to receive a constant flow rate of working liquid and an outlet section perpendicular to the axis and also having an outer surface in contact with the gas stream, such that a meniscus-forming zone exists between the outlet section of the inner surface and the discharge orifice,

the device comprising means for regulating the stagnation pressure (p0) in the convergent zone and/or means for regulating the pressure (pc) in the discharge chamber, to ensure that the speed of the gas stream downstream of the discharge orifice is equal to or greater than the speed of sound.

A final object of the present invention relates to a gaseous fluid focusing device suitable for carrying out the steps of a method according to a preferred embodiment of the first inventive aspect, the device comprising

an inner capillary, the inner capillary extending along an axis, the inner capillary having an inner surface with an inlet section configured to receive a constant flow rate of working liquid and an outlet section perpendicular to the axis and also having an outer surface,

a discharge chamber delimiting a discharge region, the discharge chamber comprising a discharge plane with a discharge orifice such that the axis of the inner capillary passes through the discharge orifice,

where

the side walls of the discharge chamber are arranged at a distance from the discharge opening of at least 3 times the width of the discharge opening, and more preferably at a distance from the discharge opening of at least 5 times the width of the discharge opening,

The discharge plane extends perpendicular to the axis of the inner capillary, so that there is a meniscus forming area between the outlet section of the inner surface of the inner capillary and the discharge orifice, and

The device comprises means for regulating the pressure (pc) downstream of the discharge orifice, to ensure that the speed of the gas stream downstream of the discharge orifice is equal to or greater than the speed of sound.

DESCRIPTION OF THE DRAWINGS

To complete the description and for the purpose of assisting a better understanding of the invention, a set of drawings is provided. These drawings form an integral part of the description and illustrate different embodiments of the present invention and data related to said embodiments, which should not be construed as restricting the scope of the invention, but as examples of how the invention can be carried out. The attached drawings specifically comprise the following figures:

Fig. 1 shows the elements comprising a first gaseous fluid focusing device.

Fig. 2 shows the elements comprising an ejector configured to implement a gaseous fluid focusing method.

Fig. 3 shows the relaxation time A* as a function of polymer concentration of two polymer solutions that can be used as working fluid in an ejector like the one in Fig. 2.

Fig. 4 shows the minimum dimensionless flow rate 0 as a function of the Weissenberg number Wi in the meniscus formation zone of an ejector like the one in Fig. 2, in the case of two polymer solutions with different concentrations.

Fig. 5 shows the shapes of the meniscus formed in an ejector like the one in Fig. 2 with different polymeric solutions, only one of them belonging to the present invention (b1), and with their respective Newtonian counterparts.

Fig. 6 shows the meniscus length and jet diameter at the discharge orifice as a function of the working liquid flow rate, for a polymeric solution according to different embodiments of the present invention, and for its corresponding Newtonian counterpart, in an ejector like that of Fig. 2.

Fig. 7 shows the diameter of the jet emitted downstream of the discharge orifice by an ejector such as that in Fig. 2, as a function of the distance from the discharge orifice (left graph), and as a function of the time of flight of the jet downstream of the discharge orifice (right graph), for two polymer solutions.

Fig. 8 shows the ratio between the length of the jet emitted by an ejector such as that in Fig. 2 and do, as a function of the ratio between the working liquid flow rate and Qo, for two polymeric solutions and different stagnation pressures in the convergent zone, according to different embodiments of the present invention.

Fig. 9 shows the minimum working liquid flow rate, the jet diameter at the discharge orifice at the minimum working liquid flow rate, and the meniscus length at the minimum working liquid flow rate, in an ejector like the one in Fig. 2, for a polymer solution as a function of polymer concentration, and for its respective Newtonian counterpart and water.

Fig. 10 shows the minimum working liquid flow rate and the jet diameter at the discharge orifice at the minimum working liquid flow rate, in an ejector like the one in Fig. 2, as a function of the polymer concentration for polymer solutions of the same polymer and different molecular weights.

Fig. 11 shows the elements comprising a second gaseous fluid focusing device different from that in Fig. 1.

Numerical references used in the figures:

DETAILED DESCRIPTION OF THE INVENTION

In this section, a description of several non-limiting examples of the method of the invention is provided. As mentioned above, the present invention relates to a method of producing highly stable, weakly viscoelastic jets by transonic gaseous fluid focusing.

Fig. 1 shows a first gaseous flow focusing (GFF) device 1 according to a preferred embodiment of the invention. Said GFF device 1 comprises a discharge chamber 2 (e.g., a glass cell) delimiting a discharge region 3 at a pressure pc (e.g., set to a pressure less than 0.1 bar by suction means, such as a pump 4), and an ejector 5 arranged inside the discharge chamber 2.

Fig. 2 shows an example of ejector 5 used in that first GFF device 1. Said ejector 5 comprises:

a convergent zone 6 at a stagnation pressure p0 comprising a receiving section 7 and a discharge orifice 8 open to the discharge region 3;

an internal capillary 9 arranged inside the convergent zone 6, said internal capillary 9 extending along an axis, and where said internal capillary 9 has an inner surface 10 with an inlet section 11 and an outlet section 12 perpendicular to the axis, and also has an outer surface 13.

Additionally, there is a meniscus forming zone 14 between the outlet section 12 of the inner surface 10 of the inner capillary 9 and the discharge orifice 8 of the convergent zone 6.

The converging zone 6 of the ejector 5 is configured to receive a gas stream 15 in the receiving section 7 and to discharge said gas stream 15 downstream of the discharge orifice 8. Here, the gas stream 15 employed is a stream of dry, impurity-free air. Also, the inlet section 11 of the inner surface 10 of the inner capillary 9 of the ejector 5 is configured to receive a working liquid 16 at a constant flow rate Q. Both the gas stream 15 and the working liquid 16 can be supplied to the ejector 5, for example, by means of valves.

On the other hand, the ratio between the stagnation pressure p0 in the convergent zone 6 and the pressure pc downstream of the discharge orifice 8 is at least that necessary for the velocity of the gas stream 15 downstream of the discharge orifice 8 to be equal to or greater than the speed of sound. With regard to the establishment of said pressures, for example, the stagnation pressure p0 in the convergent zone 6 can be established upstream with a pressure regulator of the compressed air line for positive gauge pressures, and by throttling the flow coming from the atmosphere with a valve for negative gauge pressures, while the pressure pc beyond the discharge orifice 8 can be set at a pressure lower than 0.1 bar by extracting the gas from the discharge chamber 2 using a vacuum pump 4.

Additionally, the ejector 5 is configured to implement the step of forming a meniscus 17 of working liquid 16 in the meniscus forming zone 14 when the working liquid 16 exits the internal capillary 9 and contacts the transonic gas stream 15, exerting a strain rate s- on the working liquid 16, the meniscus 17 giving rise to a stationary jet 18 of working liquid 16 that is emitted from the meniscus 17 to the discharge orifice 8.

In the ejector 5 of Fig. 2, both the inlet section 11 and the outlet section 12 of the inner surface 10 of the inner capillary 9 have the same diameter Dn, so that the working liquid 16 received at a constant flow rate Q is not accelerated within the inner capillary 9. Namely, the diameter Dn of the inner capillary 9 equals 75 .^m. On the other hand, the discharge orifice 8 has a diameter D equal to 75 .^m, and the gap H between the outlet section 12 inner surface 10 of the inner capillary 9 and the discharge orifice 8 has a size equal to 180 ^m.

Other lengths have been further illustrated in Fig. 2, which characterize the formation zone of the meniscus 14 and the stationary jet 18 of working liquid 16. These are the length lm of the meniscus 17, the diameter of the jet 18 at the discharge orifice 8, and the diameter dl of the section of the meniscus 17. Here the length lm of the meniscus 17 is defined as the distance from the outlet section 12 of the inner surface 10 of the inner capillary 9, to the section where the diameter of the jet 18 has reached eighty percent of its total reduction (Dn - de). Equivalently, the diameter dl of the section of the meniscus 17 is defined as the distance equal to the sum of twenty percent of the total reduction (Dn - de) of the diameter of the jet 18 and the diameter of the jet 18 at the discharge orifice 8.

Advantageously, the working liquid 16 of Fig. 2 is a dilute polymer solution whose polymer slightly increases the viscosity and elasticity of said polymer solution, where the elasticity is characterized by a low relaxation time A*, which grows when the concentration c of the polymer in said polymer solution increases. Hereinafter, a low relaxation time Ar refers to a relaxation time less than 30 s, and the molecular weight Mw of the polymer is less than 1x106 g/mol. Additionally, the concentration of said polymer solution is high enough for elastic stresses to exist which participate in the stabilization of the flow of working liquid 16 in the formation zone of meniscus 14. Said polymer solution can be prepared by dissolving the polymer in deionized water with a magnetic stirrer for 1 to 10 days at low angular velocities, to minimize the mechanical degradation of the polymer chains. Hereinafter, the polymer used is polyethylene oxide (PEO).

Given the conditions of Fig. 2, a deformation rate s' is exerted on the working liquid flow 16 in the meniscus formation zone 14. Said deformation rate depends on the relationship between the stagnation pressure po in the convergent zone 6 and the pressure pc downstream of the discharge orifice 8, on the geometry of the internal capillary 9 and on the flow rate Q of the working liquid 16.

Interestingly, when a polymer solution is subjected to an extensional flow that exerts on it a strain rate s', the elastic response of the polymer solution is characterized by the relaxation time Ar. Indeed, the intensity of said extensional flow to which the polymer solution is subjected can be quantified through the so-called Weissenberg number (Wi), which is a dimensionless quantity defined as Wi = A* s'. In fact, when the Weissenberg number is above 0.5, the helical-stretched transition of the dissolved polymers takes place, that is, the transition in which the polymers go from being arranged in a random helical configuration in equilibrium, to being in a stretched configuration. Likewise, the relaxation time Ar increases when the polymer concentration in a dilute polymer solution increases, due to the hydrodynamic interaction (i.e., mediated by the flow) between the polymers. Thus, for a given polymer solution, the strain rate s' required for the helical-stretch transition to take place decreases as the concentration of the polymer in the polymer solution increases.

Since the working liquid 16 is a dilute polymer solution whose viscosity is approximately constant under shear stresses, there is a first value c1* for the polymer concentration c in the polymer solution where the respective Weissenberg number is equal to 0.5 in the zone of formation of the meniscus 14.

Based on the above, if the polymer concentration c in the polymer solution is below c1* (i.e., below the viscoelastic concentration value), then its respective Weissenberg number in the meniscus formation zone 14 will be below 0.5, or in other words, the deformation rate s' exerted on the working liquid 16 in the meniscus formation zone 14 is below that required for the helical-stretching transition of the polymers in said solution to take place and, consequently, the solution will behave as a quasi-Newtonian liquid.

On the other hand, if said concentration c is equal to or greater than ci*, its respective Weissenberg number is equal to or greater than 0.5 in the meniscus formation zone 14, or what is the same, the deformation rate s' exerted on the working liquid 16 in the meniscus formation zone 14 is equal to or greater than that required for the helical-stretch transition of the polymers of said solution to take place and, consequently, elastic stresses are activated in the working liquid 16 in the meniscus formation zone 14. As a result, said elastic stresses help to stabilize the flow of working liquid 16 in the meniscus formation zone 14, and the stationary jet regime is maintained for lower working liquid 16 flow rates Q, compared to the case in which the working liquid 16 was a Newtonian liquid. And more importantly, since lower values of flow rate Q are required to produce a stationary jet 18, jets 18 with smaller diameter are obtained at the discharge orifice 8, compared to the case where the working liquid 16 was a Newtonian liquid.

Fig. 3 shows the measured relaxation time A* (in .^s, measured with an extensional rheometer) as a function of the concentration c (in wt%) for two example aqueous PEO solutions, respectively with molecular weights of 0.1*106 g/mol (PEO100K) and 0.6*106 g/mol (PEO600K). The solid lines represent the fits Ar = 26.18 c1,020 and Ar = 2707 c1,733 to the experimental results for PEO100K and PEO600K, respectively.

In the ejector 5, for example, a stagnation pressure p0 in the convergent zone 6 equal to 2.5 bar and a pressure pc downstream of the discharge orifice 8 equal to 0.075 bar can be established, and a PEO100K (PEO600K) solution can be introduced into the inlet section 11 at a flow rate Q of 0.2 ml/h (0.4 ml/h).

In this context, given the conditions set in the ejector 5 indicated above, there will be a value c1* of the concentration of PEO100K (PEO600K) for which the Weissenberg number will be equal to 0.5 in the formation zone of the meniscus 14, that is, a value ci* of the concentration for which the viscoelastic effects will be activated in the formation zone of the meniscus 14. The dashed lines in Fig. 3 indicate this value ci* and the respective relaxation time A* of the solution of PEO100K (PEO600K) at said concentration. Specifically, the value c1* of the concentration of PEO100K (PEO600K) is 0.78% by weight (0.048% by weight) and its associated relaxation time Ar is 20.3 .^s (14.0 .^s). Therefore, if in this context the concentration of these PEO solutions is equal to or exceeds the c1* values indicated by the dashed lines in Fig. 3, then the helical-stretch transition of the polymers of each solution will take place in the meniscus formation zone 14.

It should be noted that the above-mentioned working liquid flow rate Q values (0.2 ml/h for PEO100K and 0.4 ml/h for PEO600K) correspond to the minimum flow rate Qmin values compatible with the emission of a stationary jet 18 in an ejector 5 with the geometry and pressures defined above. If the same PEO solutions with the concentrations c1* of Fig. 3 were used in said ejector 5, viscoelastic effects would still be observed with higher flow rates Q, although the most advantageous situation corresponds to these minimum flow rate Qmin values, since in this case the diameter of the jet 18 at the discharge orifice 8 is the smallest possible. Moreover, these diameters are smaller than those achieved with Newtonian liquids.

Also, given the conditions set above in ejector 5, the c1* values of the PEO concentrations in Fig. 3 are not affected by using different values of stagnation pressure p0 in convergent zone 6 and pressure pc downstream of discharge orifice 8, provided that the pressure conditions set out above are met for which the velocity of the gas stream 15 downstream of discharge orifice 8 is at least the speed of sound.

Once the value c1* has been explained, there is a second value c2*, which is also relevant. This second value c2* corresponds to the concentration at which the extraction effect takes place at the outlet section 12 of the inner surface 10 of the inner capillary 9. Thus, above the second value c2* (i.e. above the extraction concentration value), the triple contact line of the meniscus 17 moves irregularly within the inner capillary 9, symmetry is lost and both the meniscus 17 and the jet 18 at the discharge orifice 8 tend to suffer from whip instabilities. Therefore, in order to avoid such instabilities, the polymer solution must have a polymer concentration c which, in addition to being greater than ci*, is lower than the second value c2*, so as to avoid the extraction effect at the outlet section 12 of the inner surface 10 of the internal capillary 9. This value c2* corresponds to a Weissenberg number equal to 10 in the meniscus formation zone 14 of the ejector 5 of Fig. 2. Specifically, given the geometric conditions of the ejector 5 defined above, and for a stagnation pressure p0 in the convergent zone 6 equal to 2.5 bar, and a pressure pc downstream of the discharge orifice 8 equal to 0.075 bar, the extraction concentration value c2* is equal to 1.13% by weight (0.071% by weight) for PEO100K (PEO600K). Curiously, and contrary to the value c1*, the value c2* may increase for lower values of the stagnation pressure p0 in the convergent zone 6. This is because, as the extraction instability results from the imbalance between the tension force originated by the Poiseuille-type flow in the internal capillary 9, and that exerted by the emitted jet 18, the variation of the stagnation pressure p0 in the convergent zone 6 affects the distribution of the hydrostatic pressure along the axis of the meniscus 17.

Fig. 4 shows the minimum dimensionless flow rate 0 as a function of the Weissenberg number Wi in the meniscus formation zone 14 of the ejector 5 in Fig. 2 for two polymer solutions of PEO100K (circles) and PEO600K (triangles) with different concentrations. The solid (for PEO100K) and dashed (for PEO600K) lines are guides for the eye. The stagnation pressure p0 in the convergent zone 6 is equal to 2.5 bar and the pressure pc downstream of the discharge orifice 8 is equal to 0.075 bar. As mentioned above, the Weissenberg number Wi in the meniscus formation zone 14 is given by the product of the relaxation time A* of the respective polymer solution by the deformation rate s' exerted on the working liquid 16 in the meniscus formation zone 14, where here said deformation rate s' is given by the quotient between the speed vj of the jet 18 near the meniscus tip 17 and the length lm of the meniscus 17.

On the other hand, the dimensionless minimum flow rate 0 is given by a piecewise function. When the concentration c of the polymer solution is below the viscoelastic concentration value c1*, the dimensionless minimum flow rate 0 is given by the Reynolds number (Re), i.e. the ratio of the minimum flow rate Qmin to the so-called characteristic minimum flow rate Q<d>of low viscosity Newtonian liquids. In this context, the latter is given by Q<d>= (Dn n)/P, where Dn is the diameter of the outlet section 12 of the inner surface 10 of the internal capillary 9 defined above, n is the viscosity of the polymer solution, and p is the density of the polymer solution. Likewise, when the concentration c of the polymer solution is above the viscoelastic concentration value c1*, the dimensionless minimum flow rate 0 is given by the capillary number (Ca), i.e. the ratio of the minimum flow rate Qmin to the so-called characteristic minimum flow rate Qv of high viscosity Newtonian liquids. In this context, the latter is given by Qv = (Dn2 a)/ He, where a is the surface tension of the polymer solution, and He is the extensional viscosity of the polymer solution, which is given by He = (3 A* a)/ dl, where dl is the diameter of the meniscus section 17 defined above.

Fig. 4 shows that the dimensionless minimum flow rate 0 is practically constant and has a similar value for both solutions when the Weissenberg number Wi takes the lowest values. In fact, this constant value of the dimensionless minimum flow rate 0 is similar to that of low viscosity Newtonian liquids. On the other hand, when the Weissenberg number Wi exceeds the value of 0.5 (i.e., when the polymer concentration c in the respective polymer solution is higher than ci*), the value of the dimensionless minimum flow rate 0 is drastically lower than for the lowest values of the Weissenberg number. This is because the helical-stretching transition takes place in the polymers of each solution in the meniscus formation zone 14 and, consequently, elastic stresses are activated which collaborate in the stabilization of the working liquid flow 16 in the meniscus formation zone 14. As long as the Weissenberg number Wi remains within the range between 0.5 and 10, the dimensionless minimum flow rate 0 increases similarly in both polymer solutions as the Weissenberg number increases, and takes values similar to those of high viscosity Newtonian liquids. In this increase, when the Weissenberg number reaches the value of 10, the value of the dimensionless minimum flow rate 0 is equal to the constant value it had for the lowest values of the Weissenberg number. At this point, the concentration c of each polymer solution is equal to its respective c2* value, that is, the extraction concentration value. Therefore, from this point on, the emitted jet 18 is prone to generate whip instabilities in the discharge channel 8.

In this context, a polymer solution with a concentration c between c1* and c2* is characterized by a notable contraction of the length lm of the meniscus 17, compared to the case in which the working liquid 16 used in the ejector 5 was a Newtonian liquid.

Fig. 5 shows the shapes of the meniscus 17 formed by an ejector 5 like that in Fig. 2 with different PEO100K solutions and with their respective Newtonian counterparts. The stagnation pressure p0 in the convergent zone 6 is equal to 2.5 bar and the pressure pc downstream of the discharge orifice 8 is equal to 0.075 bar. From here on, the term Newtonian counterpart refers to an aqueous glycerol solution (W/G) with a similar Ohnesorge number (Oh) and a flow rate Q equal to that of the respective polymeric solution. The hatched rectangles in Fig. 5 mark the inlet section 11 of the inner surface 10 of the inner capillary 9, while the white lines mark the shape of the meniscus 17.

Fig. 5 a1 (b1) shows the meniscus 17 formed with a PEO100K solution with a concentration below (above) the respective viscoelastic concentration value c1*, i.e. below (above) c1* = 0.78% by weight. At the same time, the concentrations in Fig. 5 a1-b1 are below the respective extraction concentration value c2*, i.e. c2* = 1.13% by weight. In particular, Fig. 5 a1 (b1) shows the meniscus 17 formed with a PEO100K solution with a concentration c equal to 0.5% by weight (0.875% by weight), a flow rate Q equal to 1.57 ml/h (1.38 ml/h) and an Ohnesorge number Oh equal to 0.034 (0.046). On the other hand, Fig. 5 (a2) shows the meniscus 17 formed by the Newtonian counterpart of the PEO100K solution of Fig. 5 a1 (b1), where said Newtonian counterpart has a concentration c equal to 20% by weight (28% by weight) and an Ohnesorge number Oh equal to 0.035 (0.048). In the PEO100K solution with the lowest concentration c and the lowest flow rate Q (Fig. 5 a1), a quasi-Newtonian behavior is observed, that is, the shape of the meniscus 17 formed is similar to that of its corresponding Newtonian counterpart (Fig. 5 a2). On the other hand, in the PEO100K solution of Fig. 5 b1 with higher/lower concentration/flow rate, the meniscus 17 is considerably contracted (i.e., the meniscus length lm has decreased), while the triple contact line of said meniscus 17 remains anchored to the outlet section 12 of the inner surface 10 of the inner capillary 9. Furthermore, the shape of the meniscus 17 of the PEO100K solution of Fig. 5 b1 is remarkably different from that of its corresponding Newtonian counterpart (Fig. 5 b2). Therefore, since the PEO100K and W/G solutions have a similar Ohnesorge number Oh, the contraction of the PEO100K meniscus 17 is attributed to its elasticity, i.e., said shape of the meniscus 17 is due to the weakly viscoelastic behavior of the PEO100K solution. This contraction is caused by the intense extensional flow in the zone of formation of the meniscus 14, so that the velocity of the working liquid 16 increases from values of about 1 mm/s at the outlet section 12 of the inner surface 10 of the inner capillary 9, to the velocity vj of the jet 18 of about 10 m/s near the meniscus tip 17. This sharp increase in velocity takes place over distances lm of about 100 .^m. Therefore, the resulting strain rate s' = vj/lm exerted on the working liquid 16 in the meniscus formation zone 14 is about 105 s-1, that is, above the strain rate required for the Weissenberg number to be greater than 0.5 in the meniscus formation zone 14, given the relaxation time A* corresponding to the PEO100K concentration of the solution in Fig. 5 b1 (see Fig. 3). Therefore, in Fig. 5 b1 the helical-stretch transition of the dissolved polymers takes place in the formation zone of the meniscus 14. The elastic stresses arising from this transition collaborate with those exerted by the transonic gas stream 15, pushing the polymer solution through the discharge orifice 8. The resulting additional acceleration explains the contraction of the meniscus 17. Therefore, only the polymer solution in Fig. 5 b1 forms part of the present invention.

Fig. 5 c shows the shape of the meniscus 17 of a PEO100K solution with a concentration c equal to 1.25% by weight, i.e. above the respective extraction concentration value for this solution (c2*=1.13% by weight). This is where the extraction effect is taking place. In this case, the large tensional forces arising in the meniscus 17 cause it to separate from the outlet section 12 of the inner surface 10 of the inner capillary 9, so that the triple contact line of the meniscus 17 moves upwards on the inner surface 10 of the inner capillary 9. The final position of said triple contact line is essentially determined by the balance between the tensional force originated by the Poiseuille-type flow in the inner capillary 9, and that exerted by the emitted jet 18. As a result, it can be seen in Fig. 5 c that the jet 18 tends to become unstable at the discharge orifice 8.

Advantageously, when the working liquid 16 is a polymeric solution with a concentration between c1* and c2*, the flow of working liquid 16 is stabilized in the meniscus formation zone 14 and the steady jet regime is maintained at lower flow rates Q, compared to the case where the working liquid 16 was a Newtonian liquid. Consequently, since lower flow rates Q result in the emission of a steady jet 18, smaller jet diameters are obtained at the discharge orifice 8 compared to the Newtonian case.

Fig. 6 shows the length L of the meniscus 17 and the diameter of the jet 18 at the discharge orifice 8 as a function of the flow rate Q of the working liquid 16, for a PEO100K solution according to different embodiments of the present invention, and for its corresponding Newtonian counterpart, in an ejector 5 as in Fig. 2. Specifically, the PEO100K solution has a concentration c equal to 0.875% by weight (i.e., between the corresponding c1* and c2* values) and an Ohnesorge number Oh equal to 0.046, while its Newtonian counterpart has a concentration c equal to 28% by weight and an Ohnesorge number Oh equal to 0.048. The stagnation pressure p0 in the convergent zone 6 is equal to 2.5 bar and the pressure pc downstream of the discharge orifice 8 is equal to 0.075 bar. As shown in Fig. 6, although the length lm of the meniscus 17 decreases with decreasing flow rate Q for both solutions, the length lm of the meniscus 17 is significantly shorter in the weakly viscoelastic solution, and its decrease is more pronounced than for the Newtonian counterpart. Additionally, the steady jet regime can be maintained for significantly lower flow rates Q in the weakly viscoelastic case. What happens is that, as the flow rate Q decreases, the velocity of the jet 18 near the tip of the meniscus 17 increases. Therefore, since a greater increase in the velocity of the working liquid 16 takes place over shorter distances (i.e., at shorter meniscus 17 lengths lm), the strain rate s' exerted on the working liquid 16 in the formation zone of the meniscus 14 increases. This effect intensifies the formation of elastic stresses, which explains why the contraction of the meniscus 17 increases sharply when the flow rate Q decreases. As a consequence, the flow of the working liquid 16 stabilizes at significantly lower flow rates Q, compared to the case of a Newtonian working liquid 16, i.e., lower flow rates Q are required to avoid dripping of the emitted jet 18. Furthermore, since lower flow rates Q are required, smaller diameters of the jet 18 are obtained at the discharge orifice 8 compared to the Newtonian case, as shown in the lower graph of Fig. 6.

Another additional advantage of the present invention is the increase in the stability downstream of the discharge orifice 8 of the emitted jet 18, compared to the case where the working liquid 16 was a Newtonian liquid. Thanks to said increase, the weakly viscoelastic jet breaks up at distances that are several orders of magnitude greater than those of Newtonian liquids. The reason for this stable behavior is that the elastic stresses do not relax while the weakly viscoelastic jet 18 flies downstream of the discharge orifice 8, even for times much longer than the relaxation time A* that characterizes the polymer solution that was received at the inlet section 11 of the inner surface 10 of the inner capillary 9.

Fig. 7 shows the diameter d of the jet 18 emitted downstream of the discharge orifice 8 by an ejector 5 like that in Fig. 2, as a function of the distance Z from the discharge orifice 8 (left graph), and as a function of the flight time tf of the jet 18 downstream of the discharge orifice 8 (right graph), for two PEO100K solutions. Specifically, the lines that do not end in a V-shaped arrow correspond to a jet 18 of PEO100K solution with a concentration c equal to 0.75% by weight, i.e. below its respective c1* value of viscoelastic concentration (subcritical jet), and with a flow rate Q equal to 1.6 ml/h. On the other hand, the lines ending in an arrow with a V-shaped tip correspond to a jet 18 of PEO100K solution according to the present invention, whose concentration c is equal to 1% by weight, that is, above its respective c1* of viscoelastic concentration and below its c2* value of extraction concentration (critical jet), and with a flow rate Q equal to 0.4 ml/h. The stagnation pressure p0 in the convergent zone 6 is equal to 1.1 bar and the pressure pc downstream of the discharge orifice 8 is equal to 0.015 bar. Likewise, the solutions are received in the inlet section 11 of the inner surface 10 of the internal capillary 9 with a flow rate Q just above the minimum flow rate Qmin compatible with the emission of a stationary jet 18 (see Fig. 9). The flight time tf of jet 18 is defined as the time elapsed between the instant at which the material element of jet 18 passes through discharge orifice 8, and the instant at which that element reaches the axial position Z. The solid and dashed vertical lines in Fig. 7 indicate the onset of whip (W) and Beads-on-a-String (BOAS) instabilities, respectively. BOAS-type instability is considered to be present if the diameter of the jet 18 increases by more than 10% of its mean value at that position of the jet 18. On the other hand, whip instability is considered to be present if there are whip oscillations with an amplitude greater than twice the diameter of the jet 18. Although the low pressure pc downstream of the discharge orifice 8 practically suppresses the whip instability, small amplitude lateral oscillations are observed sufficiently far from the discharge orifice 8. In any case, said amplitude in the oscillation remains small along the entire jet 18 in Fig. 7. Furthermore, the whip oscillations do not reduce the diameter of the jet 18.

The subcritical jet of Fig. 7 is progressively accelerated (the diameter decreases) downstream of the discharge orifice 8. Also, the BOAS type instability occurs at a distance Z from the discharge orifice 8 above 1 mm, while the breakup of the subcritical jet occurs at a distance Z from the discharge orifice 8 about 4 mm. On the other hand, in the critical jet of Fig. 7, the working liquid 16 undergoes a greater acceleration in the zone of formation of the meniscus 14 than in the case of the subcritical jet of Fig. 7. In fact, the diameter in the critical jet at the discharge orifice 8 is approximately the same as in the rest of the analyzed region (i.e., up to a distance Z from the discharge orifice 8 equal to 10 mm). Additionally, in the critical jet of Fig. 7, the BOAS-type instability appears at a distance Z from the discharge orifice 8 of about 4 mm, i.e., 2000 times the diameter d of the critical jet, a much larger distance than in the case of the subcritical jet of Fig. 7. Advantageously, the critical jet of Fig. 7 flows without breaking within the analyzed region, i.e., about 5000 times the diameter d of the critical jet (as indicated by the V-shaped arrow in Fig. 7). The drops formed by the BOAS-type instability are separated by distances much larger than the diameter d of the critical jet, so they do not significantly alter said diameter.

Typically, the BOAS-type instability is expected to appear at flight times tf similar to or slightly above the relaxation time A* of the polymer solution. Indeed, this is what happens for the subcritical jet in the right graph of Fig. 7. There, droplets form at times less than 100 s, while the relaxation time Ar of the polymer solution used is around 26 s. In contrast, the results for the critical jet contradict the above expectation. As can be seen in the right graph of Fig. 7, the BOAS-type instability in the critical jet is detected at flight times around 350 s, that is, at times more than one order of magnitude longer than the relaxation time of the polymer solution. This delay in the appearance of the BOAS-type instability exemplifies the improvement in the stability of the weakly viscoelastic jet of the present invention.

Fig. 8 shows the quotient between the length L of the jet 18 emitted by an ejector 5 like that of Fig. 2 and d0, as a function of the quotient between the flow rate Q of working liquid 16 and Q0, for two solutions of PEO100K (open circles) and PEO600K (filled circles), and for different stagnation pressures p0 (equal to 1, 1.1, 2.5 and 3.5 bar) in the convergent zone 6 and a pressure pc downstream of the discharge orifice 8 of 0.015 bar, according to different embodiments of the present invention. Specifically, the concentration c of the PEO100K solution is equal to 1% by weight and that of the PEO600K solution is equal to 0.058% by weight, i.e. in both cases the concentration c is above the respective c1* value of viscoelastic concentration and below the respective c2* value of extraction concentration. Here, d0 and Q0 are defined as d0 = a/p0 and Q0 = a2/(p p03)1/2, where p and a are, respectively, the density and the surface tension of the corresponding polymer solution. Additionally, a solid line is shown in Fig. 8 representing the behavior that low viscosity (Newtonian) liquids would follow, i.e., L/d0 = ap (81/2/n)3/4(Q/Q0)3/4, where ap = 15.0 is a prefactor obtained from experimental data fits, as disclosed, for example, in A. M. Gañán-Calvo et al., The natural breakup length of a steady capillary jet: Application to serial femtosecond crystallography, Crystals 11, 990 (2021). As can be seen in Fig. 8, the weakly viscoelastic jets produced in the present invention are much longer than Newtonian jets, particularly when flow rates are low. In fact, the length of the PEO100K dissolution jets in Fig. 8 becomes two orders of magnitude longer than that of Newtonian jets. Furthermore, as in Fig. 7, the PEO100K solution jet with a flow rate close to the minimum Qmin compatible with the steady jet regime does not break in the analyzed region, as indicated by the arrow in Figure 8.

In a preferred embodiment of the invention, the constant flow rate Q of working liquid 16 is set to a value 1.03 times the threshold value Qmin characterizing the beginning of the steady jet regime. Said threshold strongly depends on the concentration c of the polymer solution, and decreases drastically compared to the case of a Newtonian working liquid 16.

Fig. 9 shows the minimum flow rate Qmin compatible with a stationary jet (Fig. 9 a), the diameter dmin of the jet 18 at the discharge orifice 8 at the minimum flow rate (Fig. 9 b), and the length lm of the meniscus 17 at the minimum flow rate (Fig. 9 c), as a function of the concentration c of a PEO100K solution (circles), in an ejector 5 like the one in Fig. 2. Additionally, Fig. 9 shows the minimum flow rate Qmin compatible with a stationary jet (Fig. 9 a), the diameter dmin of the jet 18 at the discharge orifice 8 at the minimum flow rate (Fig. 9 b), and the length lm of the meniscus 17 at the minimum flow rate (Fig. 9 c), for the corresponding Newtonian counterparts (triangles). The solid lines are guides for the eye. Furthermore, the horizontal dashed lines indicate the minimum flow rate Qmin compatible with a stationary jet (Fig. 9 a), the diameter dmin of the jet 18 at the discharge orifice 8 at the minimum flow rate (Fig. 9 b), and the length lm of the meniscus 17 at the minimum flow rate (Fig. 9 c), for water. The stagnation pressure p0 in the convergent zone 6 is 2.5 bar and the pressure pc downstream of the discharge orifice is 0.075 bar. As can be seen, for concentrations c of the PEO100K solution below the corresponding c1* value (in this case, c1*= 0.78% by weight), the minimum flow rate Qmin giving rise to a stationary jet 18 increases slightly with concentration c. The same trend is observed in the Newtonian counterpart and therefore this effect is attributed to the viscosity of the polymer solution. However, when the concentration c is equal to the viscoelastic concentration value c1*, the minimum flow rate value Qmin drops and remains almost constant in the range delimited by the c1* and c2* extraction concentration values (in this case, c2*= 1.13% by weight). Within this range, the minimum flow rate value Qmin of the PEO100K solution is six times lower than that of its respective Newtonian counterpart. The sharp drop in the minimum flow rate at concentration c1* is caused by the helical-stretch transition, which triggers elastic stresses that stabilize the flow of working liquid 16 in the meniscus formation zone 14. On the other hand, the minimum flow rate value grows abruptly at concentration c equal to the c2* value. This fact is associated with the extraction effect described above. The reduction of the minimum flow rate Qmin has important practical consequences, since it allows producing finer jets 18 compared to the case of Newtonian working liquids 16 as shown in Fig. 9 b. For example, the diameter dmin at the discharge orifice 8 when the flow rate Q is set to the minimum flow rate in the case of water is approximately twice as large as when PEO100K molecules are dissolved in it at a concentration c between c1* and c2*. Interestingly, this diameter is smaller than that expected for Newtonian liquids, i.e. it is smaller than dmin ~ Qmin0.808.

Fig. 10 shows the minimum flow rate Qmin compatible with a stationary jet (upper graph) and the jet diameter dmin 18 at discharge orifice 8 at minimum flow rate (lower graph), in an ejector 5 like that in Fig. 2, as a function of polymer concentration c for PEO100K (circles), PEO200K (squares) and PEO600K (triangles) solutions. The stagnation pressure p0 in the convergent zone 6 is 2.5 bar and the pressure pc downstream of the discharge orifice is 0.075 bar. The dashed, solid and dotted lines are guides to the eye. Interestingly, although the results for the PEO200K and PEO600K dissolutions are similar to those for the PEO100K dissolution, the curves for the minimum flow rate Qmin and the jet diameter dmin 18 at the discharge orifice 8 at the minimum flow rate shift towards lower values of the concentration as the molecular weight Mw grows. This happens because the relaxation time A* grows with the molecular weight Mw and therefore, given a deformation rate s' exerted on the working liquid 16 in the meniscus formation zone 14, the viscoelastic concentration ci* value is lower the higher the molecular weight Mw. In fact, the relaxation time Ar values corresponding to the concentration ci* of PEO100K and PEO600K are of the same order (see Fig. 3). Also, when the polymer concentration c is higher than ci* in Fig. 10, the drop in the minimum flow rate value Qmin is less pronounced as the molecular weight increases. Moreover, for PEO solutions with molecular weight Mw above 106 g/mol, in an ejector 5 such as that in Fig. 2 the extraction effect is obtained before the weakly viscoelastic behaviour of the polymer solution can be observed, i.e. the extraction concentration value c2* is lower than the viscoelastic concentration value ci*.

Fig. 11 shows a second gaseous fluid focusing device 19 different from that used in Fig. 1. Said device 19 comprises:

an internal capillary 9 located in the convergent zone 6, which in this case is open to the environment, the internal capillary 9 extending along an axis, the internal capillary having an inner surface 10 with an inlet section 11 configured to receive a constant flow of working liquid 16 and an outlet section 12 perpendicular to the axis and also having an outer surface 13, and

a discharge chamber 2 (shown cut away to show its interior) delimiting a discharge region 3.

The discharge chamber 2 is cylindrical in shape with a flat upper base, which is the discharge plane, where the discharge orifice 8 is located. The side walls are arranged at a distance of 400 ^m from the discharge orifice 8, while the bottom wall is arranged at a distance of 10 mm from the discharge orifice 8, although for a better appreciation of the details of the figure, the height dimension is not shown to scale.

The convergent zone 6 is open to the environment, so it can be considered that it receives a gas stream 15 at a stagnation pressure p0 in the convergent zone 6 which is the ambient pressure (typically 1.01325 bar). The pressure pc downstream of the discharge orifice 8 is regulated by a suction pump 4 to keep it 20 times lower than the stagnation pressure p0 in the convergent zone 6 (which in this case is the ambient pressure), so that said gas stream 15 is discharged at the speed of sound.

Between the outlet section 12 of the inner surface 10 of the inner capillary 9 and the discharge orifice 8 there is a meniscus forming area 14.

In addition to the flow of the gas stream 15, the device 19 is configured to receive a working liquid 16 at a constant flow rate Q in the inlet section 11 of the inner surface 10 of the inner capillary 9.

Thus, when both streams are supplied to the device 19, a meniscus 17 of working liquid 16 is formed in the meniscus forming zone 14 when the working liquid 16 exits the internal capillary 9 and contacts the transonic gas stream 15, exerting a deformation rate s' on the working liquid 16, the meniscus 17 giving rise to a stationary jet 18 of working liquid 16 which is emitted from the meniscus 17 to the discharge orifice 8.

In the inner capillary 9 of Fig. 11, both the inlet section 11 and the outlet section 12 of the inner surface 10 of the inner capillary 9 have the same diameter Dn, so that the working liquid 16 received at a constant flow rate Q is not accelerated within the inner capillary 9. In particular, the diameter Dn of the inner capillary 9 has a size of 75 ^m. On the other hand, the diameter D of the discharge orifice 8 has a size of 75 ^m, and the gap H between the outlet section 12 of the inner surface 10 of the inner capillary 9 and the discharge orifice 8 has a size of 180 ^m.

Other dimensions are also illustrated in this Fig. 11, which characterize the meniscus formation zone 14 and the stationary jet 18 of working liquid 16, and which are analogous to those illustrated in Fig. 2 defined above.

Furthermore, the physical phenomenon carried out using a polymeric solution as defined above as the working liquid 16 is analogous to what occurs in the embodiments described based on Fig. 2, so it is not necessary to repeat all the physical mechanisms that occur in relation to this figure.

Claims (14)

1.- Method for gaseous focusing of fluids of a working liquid (16), the method comprising the stages of providing a convergent zone (6) comprising a receiving section (7) and a discharge orifice (8), the convergent zone (6) being configured to receive in the receiving section (7) a gas stream (15) and to discharge said gas stream (15) downstream of the discharge orifice (8), where the ratio between the stagnation pressure (p0) in the convergent zone (6) and the pressure (pc) downstream of the discharge orifice (8) is at least that necessary for the velocity of the gas stream (15) downstream of the discharge orifice (8) to be equal to or greater than the speed of sound, providing an internal capillary (9) within the convergent zone (6), the internal capillary (9) extending along an axis, the internal capillary (9) having an inner surface (10) with an inlet section (11) and an outlet section (12) perpendicular to the axis and also having an outer surface (13), so that there is a meniscus forming zone (14) between the inlet section (11) and the outlet section (12) perpendicular to the axis and also having an outer surface (13), so that there is a meniscus forming zone (14) between the inlet section (11) and the outlet section (12) perpendicular to the axis. outlet (12) of the inner surface (10) and the discharge hole (8), receiving a constant flow of working liquid (16) into the inlet section (11) of the internal capillary (9), forming a meniscus (17) of working liquid (16) in the meniscus forming zone (14) when the working liquid (16) exits the inner capillary (9) and contacts the gas stream (15), exerting a deformation rate on the working liquid (16), the meniscus (17) giving rise to a stationary jet (18) of working liquid (16) that is emitted from the meniscus (17) to the discharge orifice (8), the method being characterized by the working liquid (16) is a solution of a polymer with a molecular weight equal to or less than 5*106 g/mol, and more particularly, with a molecular weight equal to or less than 2*106 g/mol, and more particularly, with a molecular weight equal to or less than 1x106 g/mol, where the concentration of the polymer in the solution is between a first value and a second value, where the first value is the concentration corresponding to a Weissenberg number equal to 0.5 in the meniscus formation zone (14), and where the second value is the concentration that corresponds to a Weissenberg number equal to 10 in the meniscus formation zone (14). 2. - Method according to claim 1, wherein the working liquid (16) is received in the inlet section (11) of the internal capillary (9) at a flow rate between 1.01 and 1.1 times the minimum flow rate compatible with the emission of a stationary jet (18) of working liquid (16) from the meniscus (17) to the discharge orifice (8). 3. - Method according to any of the preceding claims, wherein the polymer is at least one of polyethylene oxide (PEO), polyvinylpyrrolidone (PVP), fibroin, a short chain peptide or any polymer that, dissolved in water in a concentration of 1% by weight, has a relaxation time equal to or less than 200 ^s, and more particularly, a relaxation time equal to or less than 100 ^s, and more particularly, a relaxation time equal to or less than 50 ^s, and more particularly, a relaxation time equal to or less than 30 ^s. 4. Method according to any of the preceding claims, wherein the gas stream (15) is an inert gas stream. 5. - Method according to any of the preceding claims, wherein the pressure (pc) downstream of the discharge orifice (8) is less than 0.1 bar, and more particularly less than 0.01 bar, and the stagnation pressure (p0) in the convergent zone (6) is greater than 10 times the pressure (pc) downstream of the discharge orifice (8), and more particularly greater than 50 times the pressure (pc) downstream of the discharge orifice (8). 6. - Method according to any of the preceding claims, wherein the discharge hole (8) has a diameter between 20 and 250 ^m, and the separation between the outlet section (12) of the inner surface (10) of the internal capillary (9) and the discharge hole (8) is between 0.5 and 5 times the diameter of the discharge hole (8). 7. - Method according to any of the preceding claims, wherein The convergent zone (6) comprises a flat plate portion, in which the discharge orifice (8) is included, so that the flat plate portion extends perpendicularly to the axis of the internal capillary (9). 8. - Method according to claim 7, wherein the convergent zone (6) is provided open to the outside, so that the stagnation pressure (p0) in the convergent zone (6) is the ambient pressure, while the region downstream of the discharge orifice (8) comprises a receptacle comprising the flat plate portion and side walls that are arranged at a distance of at least 3 times the width of the discharge orifice (8), and more preferably at a distance of at least 5 times the width of the discharge orifice (8), so that the pressure (pc) downstream of the discharge orifice (8) is regulated to achieve that the velocity of the gas stream (15) downstream of the discharge orifice (8) is equal to or greater than the speed of sound. 9. - Method for performing serial femtosecond crystallography (SFX) comprising the steps of a method according to the preceding claims, wherein the working liquid (16) comprises a sample, and the method additionally comprises the steps of irradiating with pulses of X-ray radiation the stationary jet (18) of the working liquid (16) comprising the sample, and process in a photodetector the signal resulting from the interaction between the X-ray radiation pulses and the stationary jet (18) of working liquid (16) comprising the sample. 10. - Method for producing microparticles comprising the steps of a method according to claims 1 to 8, and additionally the steps of wait for the stationary jet (18) of the working liquid (16) to break, and obtain microparticles resulting from the breaking of the stationary jet (18) of working liquid (16). 11. - Method for producing microbubbles comprising the steps of a method according to claims 1 to 8, and additionally the steps of providing one or more additional capillaries, arranged inside the inner capillary (9) and concentric to said inner capillary (9), each additional capillary extending along the same axis, each additional capillary having an inner surface with an inlet section and an outlet section perpendicular to the axis and also having an outer surface, receiving a constant flow rate of an additional working liquid at the inlet section of each additional capillary, forming microbubbles composed of each additional working liquid from each of the additional capillaries and the working liquid (16) from the inner capillary (9), when each additional working liquid exits each additional capillary and contacts the working liquid (16) exiting the inner capillary (9), wait for the stationary jet (18) of the working liquid (16) to break, and obtain the microbubbles after the breaking of the stationary jet (18) of working liquid (16). 12. - Polymeric composite obtained from the stationary jet (18) produced by a method according to any of claims 1 to 8. 13. - Gaseous fluid focusing device (1) suitable for carrying out the steps of a method according to any of claims 1 to 6, the device (1) comprising a discharge chamber (2) delimiting a discharge region (3), an ejector (5) disposed within the discharge chamber (2), the ejector (5) comprising a convergent zone (6) comprising a receiving section (7) and a discharge orifice (8) open to the discharge region (3), the convergent zone (6) being configured to receive in the receiving section (7) a gas stream (15) and to discharge said gas stream (15) downstream of the discharge orifice (8), where The ejector (5) further comprises an inner capillary (9) disposed within the discharge zone (6), the inner capillary (9) extending along an axis, the inner capillary (9) having an inner surface (10) with an inlet section (11) configured to receive a constant flow rate of working liquid (16) and an outlet section (12) perpendicular to the axis and also having an outer surface (13) in contact with the gas stream (15), such that a meniscus-forming zone (14) exists between the outlet section (12) of the inner surface (10) and the discharge orifice (8), the device (1) comprising means for regulating the stagnation pressure (p0) in the convergent zone (6) and/or means (4) for regulating the pressure (pc) in the discharge chamber (2), to ensure that the speed of the gas stream (15) downstream of the discharge orifice (8) is equal to or greater than the speed of sound. 14. - Gaseous fluid focusing device (19) suitable for carrying out the steps of a method according to claim 8, the device (19) comprising an inner capillary (9), the inner capillary (9) extending along an axis, the inner capillary (9) having an inner surface (10) with an inlet section (11) configured to receive a constant flow rate of working liquid (16) and an outlet section (12) perpendicular to the axis and also having an outer surface (13), a discharge chamber (2) delimiting a discharge region (3), the discharge chamber (2) comprising a discharge plane with a discharge orifice (8) such that the axis of the inner capillary (9) passes through the discharge orifice (8), where the side walls of the discharge chamber (2) are arranged at a distance from the discharge opening (8) of at least 3 times the width of the discharge opening (8), and more preferably at a distance from the discharge opening (8) of at least 5 times the width of the discharge opening (8), the discharge plane extends perpendicular to the axis of the inner capillary (9), so that there is a meniscus formation zone (14) between the outlet section (12) of the inner surface (10) of the inner capillary (9) and the discharge orifice (8), and The device (19) comprises means (4) for regulating the pressure (pc) downstream of the discharge orifice (8), to ensure that the speed of the gas stream (15) downstream of the discharge orifice (8) is equal to or greater than the speed of sound.