The Kessler Syndrome dispels belief so drastically that, well, the answer might just as well too.
Launch huge expandy-foamy-sticky-slime balls into earth orbit. Once in orbit, the balls would expand into gigantic foam sticky balls (made mostly of empty bubbles), maybe a few km.s in diameter. They would sweep their orbits, with all of the space debris sticking to them and becoming embedded by fibrous sticky strands that make up the foam. Eventually, the balls would accumulate so much junk inside of them, that they would become really, really heavy. They would drop out of orbit, and descend to earth. They would, of course, then burn up, taking their accumulated load of space junk with them into a fiery oblivion.
The issues that the mythology behind the Kessler Syndrome does not address is that these collisions are fission, not fusion. That is, the particles get increasingly smaller, not bigger. It is a self-limiting phenomena. Eventuality, the orbit is filled with very tiny, but very numerous, survivable particles. The second issue is that, every collision results in either a loss of velocity, or an increase in velocity. Overall, the net energy of the particles decreases. The orbits become increasingly unstable, and are erratic. The particles will have their descent into the atmosphere quickened. Third, the calculations of the energy of impact all assume that the particles end up with zero velocity, whatever that means. Particles that go right through, are defected, or just glance off do not produce anywhere near the 'released energies' quoted. Fourth, the premise ignores that fact that satellites are now 'hardened' against expected impacts. They are able to absorb the impact energies. And lastly, the problem is minuscule compared to how much space junk hits earth.
60 Tons Of Cosmic Dust Fall To Earth Every Day
Until now, scientists didn't know how much of this cosmic dust was
gathering on Earth (though they know rather a lot about how much is up
in space). Researchers guessed that anywhere between 0.4 and 110 tons
of the star stuff entered our atmosphere every day--that's a pretty
wide range. But a recent paper took a closer look at the levels of
sodium and iron in the atmosphere using Doppler Lidar, an instrument
that can measure changes in the composition of the atmosphere. Because
the amount of sodium in the atmosphere is proportional to the amount
of cosmic dust in the atmosphere, the researchers figured out that the
actual amount of dust falling to the earth is along the lines of 60
tons per day.
The problem is analogous to the 'Y2K' problem that was hyped at the turn of the century. It had some factual merit, but please, airplanes falling out of the sky on Jan.1,2000? Over-hyped by far.
What is missing from the question, especially for it to be science-based, or even hard science, are any criteria. So, here are a few potential criteria. The solution, of course, could not make the problem worse, by producing even more stuff. Second, the solution would have to be capable of handling or dissipating the potential and kinetic energy of any 'captured' particle, so that no further destruction or damage is done. Third, by far the greatest problem is junk that is smaller than one cm. Any solution that does not handle this should be a non-starter. And fourthly, the solution should be compliant with existing knowledge and information about the problem.
The mean number of fragmentations per year accounts to 4.9 (SD = 2.8)
with no sign of regression or fundamental change. However, there was a
slight decrease in fragmentations for three years after the
introduction of the Inter-Agency Space Debris Coordination Committee
(IADC) Space Debris Mitigation Guidelines in 2010. An examination of
the fragmented mass over the years is shown in Figure 2b. It is shown
that most of the objects that experienced a breakup had a mass less
than 500 kg. Looking at the other extreme of the histogram, there are
two events with a fragmentation mass of 26,000 kg and 30,000 kg. These
events happened in the late 1960s during the Apollo program, but, due
to their low event altitude of below 300 kg, almost all fragments
re-entered into Earth’s atmosphere and consequently do not contribute
to the space debris environment in the long term.
So a huge ball of space foam would be ideal for trapping these small particles. The volume would be great enough to cause slow deceleration, any small particles that were released would also be trapped, the ball is big enough to be visible, and thus not a navigation hazard, and it is self-healing. Larger particles would just blow through it, and the gap would close.
However, such a material and such a method does not currently exist. The foam could never 'solidify', as it would then be subject to fracturing and thus contributing to the problem. The foam would also have to be sticky, and it would also have to consist of long polymers, such that they would bind and hold together, but also be able to slide along each other, so as to absorb the collision energy. Thus, the answer is just as conjectural as the question.